1
Two-Dimensional Photonic Crystals for Engineering Atom-Light Interactions
Su-Peng Yu*,
1
Juan A. Muniz*,
1
Chen-Lung Hung,
2, 3
and H. J. Kimble
1
1)
Norman Bridge Laboratory of Physics MC12-33, California Institute of Technology
Pasadena, California 91125, USA
2)
Department of Physics and Astronomy, Purdue University,
West Lafayette, Indiana 47907, USA
3)
Purdue Quantum Center, Purdue University,
West Lafayette, Indiana 47907, USA
(Dated: 24 December 2018)
We present a two-dimensional (2D) photonic crystal system for interacting with cold cesium (Cs) atoms. The
band structures of the 2D photonic crystals are predicted to produce unconventional atom-light interaction
behaviors, including anisotropic emission, suppressed spontaneous decay and photon mediated atom-atom
interactions controlled by the position of the atomic array relative to the photonic crystal. An optical
conveyor technique is presented for continuously loading atoms into the desired trapping positions with
optimal coupling to the photonic crystal. The device configuration also enables application of optical tweezers
for controlled placement of atoms. Devices can be fabricated reliably from a 200nm silicon nitride device
layer using a lithography-based process, producing predicted optical properties in transmission and reflection
measurements. These 2D photonic crystal devices can be readily deployed to experiments for many-body
physics with neutral atoms, and engineering of exotic quantum matter.
Keywords: Nanophotonics
|
Quantum optics
|
Quantum Many-Body
Specialized two-dimensional photonic crystals
have been developed to interact with ultra-cold
atoms, which are identical particles demonstrat-
ing quantum behavior both in their interac-
tion with photons and in their motional degrees
of freedoms. In the system presented here,
the quantum nature of atoms is complemented
with capabilities of 2D photonic crystals to en-
gineer 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 new tools to engineer
quantum many-body systems and create exotic
quantum matters.
The introduction of nano-photonics to the field of
quantum optics and atomic physics greatly broadens
the capabilities of atom-photon systems
1
.
Interac-
tion with photonic structures such as tapered optical
fibers
2–5
, micro-cavities
6–8
, and photonic crystal (PhC)
waveguides
9–11
enable engineering of atom optical prop-
erties 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
12–14
,
color centers
15,16
, and embedded rare-earth ions
17–19
.
Among the variety of quantum emitters available nowa-
days, the identical-particle nature of neutral atoms pro-
vides particular advantages in forming quantum many-
1
S.-P. Yu and J. A. Muniz contributed equally to this work
body systems. Both the center-of-mass motion of the
atoms and the photons they interact with can potentially
become components of such quantum systems
20,21
, and
interaction between neutral atoms can be introduced by
coupling them to shared photonic modes
22,23
. Recently,
there is an interest on exploring how 2D arrays of cold
atoms coupled to a photonic bath show interesting topo-
logical properties robust under scattering
24,25
. The array
of capabilities of atom-optics systems provide building
blocks for exotic quantum many-body systems.
The cold atom community has put significant efforts
into creating controllable interacting 2D systems
26
. Ad-
vances in optical lattice systems allow for creation of
various geometries of lattices
27,28
, engineering of many-
body Hamiltonians
29
, and manipulating atoms on a sin-
gle site
30
. The 2D platforms demonstrate physics that
is not manifest in 1D geometries, such as frustrated spin
systems
31
and directional emission
32,33
.
It is interesting to explore beyond the photonic cavity
and 1D waveguide systems, and create equally versatile
2D photonic systems. The particular platform of inter-
est here is the 2D photonic crystal slab, where a periodic
spatial modulation in dielectric distribution is arranged
over a dielectric slab. The properties of a photonic crystal
can be exploited to provide sub-wavelength scale optical
trapping and dispersion engineering
22,34,35
. In this arti-
cle, we present two distinct types of 2D photonic crystal
slabs demonstrating novel photonic properties, such as
directional spontaneous emission, spontaneous emission
suppression, engineering of coherent atom-atom interac-
tions, and novel topological properties in linear dielectric
systems. Optical trapping schemes for trapping neutral
ultra-cold Cs atoms in the vicinity of the photonic crystal
arXiv:1812.08936v1 [physics.atom-ph] 21 Dec 2018
2
slabs are also devised.
The 2D photonic crystal slabs are fabricated using elec-
tron beam lithography and standard etching processes,
as reported in
36
, with a single suspended device layer
of silicon nitride (SiN). They provide sufficient optical
access to enable application of laser cooling and trap-
ping techniques, such as magneto optical trap (MOT)
and optical dipole trap in close vicinity of the photonic
crystal structures. An overview of our photonic crystal
devices is shown in Fig. 1(a), where on-chip waveguides
are connected to conventional optical fibers for efficient
addressing to the guided modes of the 2D PhC slab by
means of a self-collimation scheme
35,37,38
. This enables
direct optical characterization of their properties. The
two types of lattice structures to be presented here are:
a square lattice of circular holes, as shown in Fig. 1(b),
where we explore the anisotropic emission of excited state
atoms into the guided photonic modes; and a triangular
array of hexagonal holes, depicted in Fig. 1(c), where we
will focus on configurations with atomic resonances in
the photonic band-gap.
25
μ
m
a)
b)
c)
100 nm
100 nm
FIG. 1. A suspended 2D PhC slab. (a) SEM image of a pho-
tonic crystal slab structure, suspended by two single-beam
SiN waveguides on left and right edges. The slab contains
multiple sections with smoothly varying crystal parameters
for guiding purposes (see Fig. 6 and discussions). The dielec-
tric 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) Zoom-in SEM image of
the square lattice of circular holes and (c) triangular lattice
of hexagonal holes photonic crystals.
ENGINEERING ATOM-PHOTON INTERACTIONS
The two photonic crystals slab structures display dif-
ferent regimes for light-matter interactions. For both de-
signed 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 op-
tical properties such as flat landscape of group veloci-
ties and opening of photonic band-gaps, while satisfying
practical requirements such as minimum feature sizes and
mechanical robustness.
The classical electromagnetic
Green’s tensor is then calculated using Finite-Difference
Time-Domain (FDTD) methods on a simulated finite-
size photonic crystal slab. Finally, further properties
such as the emission characteristics of a dipole near the
crystal can be obtained from the Green’s tensor
39–41
.
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
39,41–43
Γ
Total
=
8
π
2
μ
0
̄
h
·
∑
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 consideration, and the summation
is over all ground states
|
g
i
〉
. In order 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 crys-
tal, instead of emitting into free-space or other loss chan-
nels. Such performance can be characterized by the ratio
Γ
2D
/
Γ
′
, where Γ
2D
,
Γ
′
are the atom decay rate into the
guided modes of interest, and into any other channels, re-
spectively, 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 study-
ing 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 symme-
try of the dielectric pattern. If the dipole frequency lies
outside a band-gap and thus radiates into the propagat-
ing guided modes, highly anisotropic emission can be ob-
served over select frequency ranges
32,33,44
. Furthermore,
the vector character of the guided mode electric field and
the tensor components of the atomic electric dipole op-
erator can further affect the spatial emission pattern.
Our structures were designed to engineer interactions
between Cs atoms and the transverse-electric (TE)-like
3
250
350
450
150
50
M
X
Y
a
R
x
y
X
M
Y
Γ
X
Γ
M
a)
b)
d)
e)
c)
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
k
Γ
M
FIG. 2. Properties of 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. The real space (black arrows) and momentum space (red arrows) basis vectors are shown in the
inset. (b) Equi-frequencies curves (EFC) 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)-(d)-(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 is indicated by the blue arrows. In (c), the dipole frequency is
ν
= 320 THz, 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.
guided modes, where the electric field is polarized pre-
dominately 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 utilized for optical trapping and
studying light-matter interactions. The crystal dimen-
sions are chosen such that the frequencies at high sym-
metry points are aligned to the Cs D
1
and/or D
2
tran-
sitions at 335 THz (894 nm) and 351 THz (852 nm),
respectively. 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.
A. Anisotropic spontaneous emission in a square lattice
photonic crystal slab
We first consider a PhC slab consisting of a square lat-
tice 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 pattern-
ing on the PhC as seen in Fig. 2(b), which manifest as
equi-frequency curves (EFC) of constant guided mode
4
10
10
10
-1
1
0
300
340
380
Γ
K
Γ
M
200
400
a
t/2
x
y
M
K
x
y
z
a)
b)
c)
A
A
J
J
J
J
J
J
A
A
=
= m
a
J = J = J
A’
= (m+1)
a
y
J >0, J =0, J<0
~
A
A
J
J
J
J
J
J
A
A
A’
=
+
A
J > J = J
d)
e)
f)
g)
-2
-1
0
1
2
2
1
0
-1
-2
-1
0
1
x (
μ
m)
y (
μ
m)
x, y
z
A
A
a
D
1
D
2
Re(
T
+1
)
a = 405nm
FIG. 3. Properties 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 (shaded horizontal region) 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 (see 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). The excitation dipole position and polarization are shown in the
inset. (c) The real part of the Green’s tensor component Re(T
+1
(
~r,ν
0
)), normalized by setting
|
Re(T
+1
(
~r,ν
0
))
|
at the nearest-
neighbor cell from the dipole position (
~r
0
) to 1. 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 super-lattice of
atoms associated with lattice vectors
~
A
1
,
~
A
2
is formed by placing atoms in selected sites in the photonic crystal. The following
interaction parameters
J
i
can be engineered: (e)
J
1
=
J
2
=
J
3
, with
J
i
=
−
0
.
22
<
0 for
m
11
=
m
22
=
m
= 3, and
J
i
= 0
.
06
>
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
.
06
,J
2
=
−
0
.
01
,J
3
=
−
0
.
06,
which has a weak coupling to the
J
2
site, forming an effective square-lattice-like interaction; and (g) stretching the super-lattice
perpendicularly to
~
A
1
from the m=2 case creates an anisotropic interaction of
J
1
= 0
.
72 while
J
2
=
J
3
= 0
.
14
.
frequencies. The group velocity
~v
g
= 2
π
∇
~
k
ν
(
~
k
) is per-
pendicular 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, in-
dicating that the group velocity points in the same direc-
tion (i.e., along
~
k
Γ
M
as in Fig. 2(b)). Therefore, all exci-
tations with those wave-vectors propagate approximately
in the same direction. This gives rise to a self-collimation
effect
34,45
that leads to directional emission
32,33,44,46,47
.
In Fig. 2(c-e) we study the effect of engineered band
structure on the dipole emission pattern, using the EFC
as a guidance. We consider a dipole placed at the center
of a hole, polarized parallel to the slab. For a dipole radi-
ating at
ν
= 320 THz and polarized along the
y
direction,
as in Fig. 2(c), the radiation pattern is roughly isotropic
and no preferred direction is found. However, if the ra-
diation 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).
B. Triangular lattice photonic crystal slab
A photonic crystal with a triangular lattice of hexago-
nal holes was designed to create a band-gap for the TE-
like modes. As indicated in Fig. 3(a), we parametrize the
photonic crystal unit cell by its lattice constant
a
, and
the width
t
of the dielectric tether separating adjacent
holes. The unit cell geometry 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 calculating
the Green’s tensor in the hole center of a unit cell of a
photonic crystal slab; see 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 polarizations, a suppression of up to 8dB on
the spontaneous emission rate can be achieved. There
is no significant suppression of decay for the polariza-
tion perpendicular to the device plane in the frequency
range being considered. This behavior is not reproduced
in structures such as the square lattice of holes discussed
before, where there is not a complete TE-like band-gap.