Current affiliation: ]Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS,
Grenoble INP, IMEP-LAHC, 38000 Grenoble, France.

# Asymmetric comb waveguide for strong interactions between atoms and light

N. Fayard  Université Paris-Saclay, Institut d’Optique Graduate School, CNRS,
Laboratoire Charles Fabry, 91127 Palaiseau, France  A. Bouscal  Laboratoire
Kastler Brossel, Sorbonne Université, CNRS, ENS-Université PSL, Collège de
France, 4 place Jussieu, 75005 Paris, France  J. Berroir  Laboratoire Kastler
Brossel, Sorbonne Université, CNRS, ENS-Université PSL, Collège de France, 4
place Jussieu, 75005 Paris, France  A. Urvoy  Laboratoire Kastler Brossel,
Sorbonne Université, CNRS, ENS-Université PSL, Collège de France, 4 place
Jussieu, 75005 Paris, France  T. Ray  Laboratoire Kastler Brossel, Sorbonne
Université, CNRS, ENS-Université PSL, Collège de France, 4 place Jussieu,
75005 Paris, France  S. Mahapatra  Centre de Nanosciences et de
Nanotechnologies, CNRS, Université Paris-Saclay, 91120 Palaiseau, France  M.
Kemiche  [  Centre de Nanosciences et de Nanotechnologies, CNRS, Université
Paris-Saclay, 91120 Palaiseau, France  J. A. Levenson  Centre de Nanosciences
et de Nanotechnologies, CNRS, Université Paris-Saclay, 91120 Palaiseau, France
J.-J. Greffet  Université Paris-Saclay, Institut d’Optique Graduate School,
CNRS, Laboratoire Charles Fabry, 91127 Palaiseau, France  K. Bencheikh  Centre
de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay, 91120
Palaiseau, France  J. Laurat  Laboratoire Kastler Brossel, Sorbonne
Université, CNRS, ENS-Université PSL, Collège de France, 4 place Jussieu,
75005 Paris, France  C. Sauvan<EMAIL_ADDRESS>Université Paris-Saclay, Institut d’Optique Graduate School, CNRS, Laboratoire
Charles Fabry, 91127 Palaiseau, France

###### Abstract

Coupling quantum emitters and nanostructures, in particular cold atoms and
waveguides, has recently raised a large interest due to unprecedented
possibilities of engineering light-matter interactions. However, the
implementation of these promising concepts has been hampered by various
theoretical and experimental issues. In this work, we propose a new type of
periodic dielectric waveguide that provides strong interactions between atoms
and guided photons with an unusual dispersion. We design an asymmetric comb
waveguide that supports a slow mode with a quartic (instead of quadratic)
dispersion and an electric field that extends far into the air cladding for an
optimal interaction with atoms. We compute the optical trapping potential
formed with two guided modes at frequencies detuned from the atomic
transition. We show that cold Rubidium atoms can be trapped as close as 100 nm
from the structure in a 1.3-mK-deep potential well. For atoms trapped at this
position, the emission into guided photons is largely favored, with a beta
factor as high as 0.88 and a radiative decay rate into the slow mode 10 times
larger than the free-space decay rate.

##  I Introduction

The development of quantum technologies requires a number of prerequisites
among which strong atom-photon interactions hold a high rank kimble2008quantum
; chang2014quantum . Enhancing the interaction between a single photon and a
single atom has been a driving force for a large community over the past
decades. Two main routes have first been followed rather independently: atoms
in macroscopic high-finesse cavities raimond2001manipulating ;
reiserer2015cavity  or solid-state emitters (e.g., quantum dots) in micro and
nanostructures GerardPRL98 ; lodahl2015interfacing . In recent years, various
works have combined both approaches by interfacing cold atoms with
nanophotonic devices such as photonic crystal nanocavities
tiecke2014nanophotonic , nanofibers nayak2007optical ; vetsch ;
goban2012demonstration , and photonic crystal waveguides goban2014atom ;
goban2015superradiance ; zang2016interaction . These hybrid strategies benefit
from both the long coherence time of atoms and the enhanced electromagnetic
field associated with subwavelength light confinement in nanostructures.

Besides the well-established field of cavity quantum electrodynamics (QED),
the use of single-pass schemes with nanofibers or nanostructured waveguides
has triggered the emergence of a new field of research known as waveguide QED
roy2017colloquium ; chang2018colloquium ; turschmann2019coherent . Photons
travelling in the waveguide carry the information through long distances while
atoms can store it for long times. These systems are a promising building
block for quantum networks as shown recently by the experimental
demonstrations of a coherent photon storage gouraud2015demonstration ;
sayrin2015storage , the heralded creation of a single collective excitation of
atomic arrays corzo2019waveguide , or a correlated photon transport
mahmoodian2018strongly ; mahmoodian2020dynamics ; prasad2020correlating . In
addition, strong atom-photon interactions give rise to new phenomena in many-
body physics douglas2015quantum  such as the emergence of solitons dynamics
calajo2021few ; bakkensen2021photonic  or many-body localization
fayard2021many .

Most of the existing experiments in waveguide QED at optical frequencies are
performed with silica nanofibers nayak2007optical ; vetsch ;
goban2012demonstration ; gouraud2015demonstration ; sayrin2015storage ;
sorensen2016coherent ; Corzo2016 ; solano2017super ; corzo2019waveguide ;
prasad2020correlating . They offer a 4$\pi$ solid-angle access to the
interaction region that markedly eases the manipulation of atomic clouds near
the structure. In addition, nanofibers provide single-mode operation over a
broad spectral range. It is thus possible to use guided light beams at
frequencies detuned from the atomic transition for trapping the atoms at
subwavelength distances from the waveguide goban2012demonstration ;
balykin2004atom ; le2004atom . However, the atom-photon interaction remains
relatively weak. It can be quantified by the $\beta$ factor, which is defined
as the ratio of the radiative decay rate of a single atom in the waveguide
mode $\Gamma_{1\textrm{D}}$ to the total decay rate $\Gamma_{\textrm{tot}}$,
$\beta=\Gamma_{1\textrm{D}}/\Gamma_{\textrm{tot}}=\Gamma_{1\textrm{D}}/(\Gamma_{1\textrm{D}}+\Gamma^{\prime})$.
Photons that are not funnelled into the waveguide mode are lost in the
radiation continuum with a decay rate $\Gamma^{\prime}$. Experimentally
observed values of $\beta$ in nanofibers are typically in the range of
$\beta\sim 0.05$ nieddu2016optical ; forn2019ultrastrong ;
sheremet2021waveguide .

A promising route to increase $\Gamma_{1\textrm{D}}$ and $\beta$ is to use
periodic dielectric waveguides, i.e., waveguides with a periodic modulation of
the refractive index along the propagation direction. Indeed, the radiative
decay rate of a single atom in a guided mode is given by
$\Gamma_{1\textrm{D}}/\Gamma_{0}=n_{g}\sigma/(2A_{\textrm{eff}})$, where
$\Gamma_{0}$ is the atomic decay rate in free space, $\sigma$ is the
absorption cross-section, $n_{g}$ and $A_{\textrm{eff}}$ are the group index
of the mode and the effective area at the atom position goban2014atom ;
lecamp2007prl . In periodic waveguides, the coupling between contrapropagating
modes results in the opening of bandgaps in the dispersion relation and in the
apparition of band edges where the group velocity $v_{g}=c/n_{g}$ goes to zero
joannopoulos2008molding . Close to these peculiar points, periodic waveguides
support slow guided modes with large $n_{g}$’s, which, in turn, lead to
increased values of $\Gamma_{1\textrm{D}}$ and $\beta$ lecamp2007prl . Hence,
an important research effort has been devoted to the design and the
realization of periodic waveguides aimed at increasing the interaction with
cold atoms goban2014atom ; goban2015superradiance ; zang2016interaction ;
hung2013trapped ; yu2014nanowire . However, this is a challenging task that
requires tackling various issues zang2016interaction ; kimble2020optica ,
including the sensitivity of slow light to fabrication imperfections, a stable
trapping of atoms at the desired positions, and the accessibility around the
structure to ease the transport of the atoms to the trapping sites.

In addition to enhanced atom-photon interactions, periodic waveguides also
offer the possibility to create dispersion relations that greatly differ from
the usual and almost linear dispersion of a nanofiber. The curvature of the
dispersion relation between the frequency $\omega$ and the wavevector $k$ has
a profound impact on various physical phenomena, such as the distortion of
short pulses during propagation, the collective properties of atomic arrays
coupled to the waveguide, the localization of photons emitted in the bandgap
chang2018colloquium , as well as the tolerance of the slow mode to inevitable
fabrication imperfections zang2016interaction ; FaggianiSR16 . However, the
engineering of the dispersion has been quite limited until now. Indeed, most
of the periodic waveguides studied so far (be it for interacting with atoms or
for another purpose) are made of a periodic pattern that is symmetric in
transverse directions goban2014atom ; goban2015superradiance ; hung2013trapped
; joannopoulos2008molding ; LoncarAPL00 ; NotomiPRL01 ; DeLaRueOE09 ;
ChebenOE10 ; ChebenOE12 ; ChebenOL14 ; VivienSR19  and their dispersion
relation varies most often quadratically at band edges.

By breaking the transverse symmetry, one can obtain new degrees of freedom for
engineering the dispersion beyond the standard parabolic shape. Recently,
Nguyen _et al._ demonstrated with a two-dimensional (2D) structure that
symmetry breaking can be used to create exotic photonic dispersion relations
such as Dirac cones, multivalley curves, or flat bands nguyen2018symmetry .

Figure 1: Scheme of the proposed asymmetric comb waveguide. The structure is
etched in a GaInP membrane (refractive index $n=3.31$) suspended in air. The
waveguide has a total width $H$ in the transverse $x$ direction and a
thickness $t_{y}$ in the transverse $y$ direction. The comb pattern along the
propagation direction $z$ is made of teeth with a width $w$ and a height
$H_{\textrm{etched}}$, periodically spaced with a period $a$. The width of the
guiding region is thus $H-H_{\textrm{etched}}$. An excited atom (red dot)
decays radiatively either into the guided mode, with a rate
$\Gamma_{1\textrm{D}}$, or into the radiation continuum with a rate
$\Gamma^{\prime}$.

In this article, we exploit symmetry breaking to design a novel three-
dimensional (3D) geometry, the asymmetric comb waveguide, that (i) supports a
slow mode with a _quartic_ dispersion, (ii) offers the possibility to trap
atoms optically at subwavelength distances, and (iii) provides very large
$\beta$ factors for trapped atoms. We show that the quartic dispersion makes
the slow mode more tolerant to fabrication imperfections than previous
proposals zang2016interaction . The structure is represented in Fig. 1. It
consists of a suspended bridge waveguide (with a 4$\pi$ solid-angle access to
the interaction region like a nanofiber) of width $H$ and thickness $t_{y}$
that has been periodically corrugated with an asymmetric rectangular pattern.
This promising structure that provides both a strong atom-photon interaction
and a dispersion that differs from the usual parabolic shape opens new
opportunities in waveguide QED.

The article is organized as follows. In Section II, we illustrate with 2D
examples the role of symmetry breaking in the design of a flat band with a
quartic dispersion. Then, we design in Section III a 3D comb waveguide with a
quartic dispersion after having detailed a few important criteria that should
be fulfilled by a periodic waveguide aimed at interacting with cold atoms. We
present in Section IV the calculation of the potential of a two-color optical
trap made with guided modes. We show that cold Rubidium (Rb) atoms can be
trapped as close as 100 nm from the waveguide in a 1.3-mK-deep potential well
by using relatively low powers ($P\sim 1$ mW) compatible with photonic
structures. Finally, we calculate in Section V the emission rates
$\Gamma_{\textrm{1D}}$, $\Gamma_{\textrm{tot}}$, and $\Gamma^{\prime}$ of a
trapped atom. We provide evidence for very large coupling between the atom and
a slow mode with $n_{g}=50$: $\Gamma_{\textrm{1D}}=10\Gamma_{0}$,
$\Gamma^{\prime}=1.3\Gamma_{0}$, and $\beta=0.88$. Section VI concludes the
work and discusses some perspectives.

##  II Slow mode with a quartic dispersion

In this Section, we show that symmetry breaking can be used to create a comb
waveguide that supports a slow mode with a quartic dispersion around a zero-
group-velocity point instead of the usual parabolic shape. In order to
illustrate the impact of symmetry breaking, we compare the band diagrams of
four different periodic waveguides with a period $a$. The waveguides are
schematically represented in the insets of Figs. 2(a)-(d). All four are comb
waveguides (see Fig. 1) with the same total width $H=2a$ but different
corrugations. The comparison is done with 2D structures ($t_{y}=\infty$) in TM
polarization (magnetic field polarized in the $y$ direction). We consider a
refractive index $n=2.85$ that corresponds to the effective index of the
fundamental guided mode in a GaInP membrane of thickness 150 nm at
$\lambda=780$ nm.

The dispersion curves are calculated with a Bloch-mode solver developed for
studying light scattering in periodic waveguides LecampOE07 . The solver
calculates the wavevector $k$ in the propagation direction as a function of
the frequency $\omega$. It is implemented with the aperiodic Fourier modal
method (a-FMM) Silberstein01 , which relies on an analytical integration of
Maxwell’s equations along the direction of periodicity ($z$ axis) and on a
supercell approach with perfectly-matched layers (PMLs) along the transverse
directions ($x$ and $y$ axis). PMLs ensure a correct treatment of far-field
radiation conditions; they are implemented as complex nonlinear coordinate
transforms Hugonin05 . This numerical method is also used in the rest of the
paper for the calculation of the field distributions and the decay rates.

Figure 2: Role of a transverse symmetry breaking. (a)-(d) Band diagram
$\omega=f(k)$ of four different 2D comb waveguides in TM polarization. The
total height of all structures is $H=2a$, with $a$ the period. To bring these
2D calculations closer to 3D simulations, we consider an effective refractive
index $n=2.85$. (a) Symmetric comb waveguide. The width of the teeth on both
sides is $w=0.5a$ and their depth is $H_{\textrm{etched}}=0.25H$. (b)
Asymmetric comb waveguide. The teeth are two times deeper,
$H_{\textrm{etched}}=0.5H$, with the same width $w=0.5a$ to conserve the
fraction of matter. (c) Asymmetric comb waveguide with a sinusoidal
corrugation but the same fraction of matter. The depth of the cosine
modulation is chosen so that the quantity of matter is the same as in (a)-(b).
(d) Asymmetric comb waveguide designed to support a flat band with a quartic
dispersion (green curve). The width of the teeth is $w=0.372a$ and their depth
is $H_{\textrm{etched}}=0.8H$. (e) Variation of
$\Delta\omega=|\omega-\omega_{e}|$ as a function of $\Delta k=|k-\pi/a|$ in
logarithmic scales for the green band in (d) and the red band of lowest
frequency in (a). We observe that $\Delta\omega\sim\Delta k^{4}$ for the flat
band of the asymmetric comb and $\Delta\omega\sim\Delta k^{2}$ for the band of
the symmetric comb. Those different behaviors lead to different scaling laws
for the variation of the group index as a function of
$\Delta\lambda=|\lambda-\lambda_{e}|$ as shown in (f).

Figure 2(a) displays the band diagram of a comb waveguide whose corrugation is
symmetric in the transverse direction. It can be easily understood with the
following picture: the uncorrugated planar waveguide of width $H$ supports
symmetric and antisymmetric modes and the periodic corrugation couples forward
and backward propagating modes, resulting in the opening of photonic bandgaps
joannopoulos2008molding . However, the symmetry of the corrugation forbids
coupling between modes of different symmetry. Therefore, the Bloch modes
resulting from the coupling can be sorted in two distinct families: they are
either symmetric (blue curves) or antisymmetric (red curves). As can be seen,
blue and red curves cross each other without coupling. Most of the bandgaps
and the associated zero-group-velocity points are located at the edge of the
first Brillouin zone, at $k=\pi/a$. Using a standard coupled-mode approach
with two waves having linear dispersions, one can show that the dispersion
relation of the Bloch modes varies quadratically in the vicinity of these
points: $\omega-\omega_{e}\propto\pm(k-\pi/a)^{2}$, where $\omega_{e}$ is the
frequency of the band edge where $v_{g}=0$.

We now consider in Figs. 2(b)-(c) comb waveguides without mirror symmetry in
the transverse direction and with two different corrugations profiles. For the
sake of comparison, the fraction of matter (the orange area) is the same as in
Fig. 2(a). The situation is now fundamentally different: since the
corrugations have no particular symmetry, all possible couplings are indeed
allowed. As a result, gaps are now open inside the Brillouin zone
($k\neq\pi/a$) around the points of intersection between the dispersion curves
of the symmetric and antisymmetric modes of the uncorrugated waveguide.
Looking precisely to the band diagrams, we observe that the spectral widths of
the bandgaps are smaller for a sinusoidal profile than for a rectangular
profile. Thus, the latter provides larger coupling strengths. This is the
reason why we study the rectangular asymmetric comb waveguide in the
following.

For a rectangular profile, the coupling strengths depend on the size of the
teeth, fixed by the width $w$ and the depth $H_{\textrm{etched}}$ defined in
Fig. 1. Stronger corrugations result in larger coupling strengths and wider
bandgaps. If the coupling is strong enough, two consecutive bandgaps may
overlap. In this case, the coupling mechanism is more complex since it
involves four waves instead of two nguyen2018symmetry . This offers extra
degrees of freedom for engineering the bands beyond the standard parabolic
shape.

Following this principle, we have designed an asymmetric comb waveguide that
supports a flat band with a _quartic_ dispersion curve of the form
$\omega-\omega_{e}\propto-(k-\pi/a)^{4}$. This flat band is highlighted in
green in Fig. 2(d). In order to confirm the quartic variation, we plot in Fig.
2(e) $\Delta\omega=|\omega-\omega_{e}|$ as a function of $\Delta k=|k-\pi/a|$
with logarithmic scales. The solid green curve that is extracted from the
green band in Fig. 2(d) follows a clear quartic variation, as indicated by the
dashed green line. In contrast, the symmetric comb waveguide of Fig. 2(a)
supports modes with a quadratic dispersion, as shown by the solid and dashed
red curves.

A flat band with a quartic dispersion broadens the useful bandwidth of the
slow mode, i.e., the bandwidth over which $n_{g}$ is larger than a target
value. Indeed, a quartic dispersion produces a group index that scales with
the wavelength as $\Delta\lambda^{-3/4}$ while the group index of a quadratic
dispersion scales as $\Delta\lambda^{-1/2}$, as can be seen in Fig. 2(f).
Therefore, if one wants to work at a given group index, the quartic dispersion
(green curve) allows an operation at a larger $\Delta\lambda$, i.e., at a
frequency further from the band edge. We will see in the next Section that
this increase of the useful bandwidth is important to improve the tolerance of
the slow mode to fabrication imperfections.

Let us end this Section by emphasizing that a quartic functional form for the
dispersion relation of a slow mode is uncommon, in particular in the context
of waveguide QED where it has not been studied so far. It is thus important to
design a waveguide with a quartic dispersion and a 3D geometry compatible with
the interaction with cold atoms trapped in the air cladding. We present this
design in the following Section.

##  III Design of a three-dimensional asymmetric comb waveguide

We first detail a few important requirements that apply to a slow-light
waveguide designed to interact with cold atoms. Then, we present the design of
a 3D asymmetric comb waveguide that supports a slow mode with a quartic
dispersion at the transition frequency of Rb atoms and an evanescent field
that extends far into the air cladding for an optimal interaction. In
addition, we pay particular attention during the design process to the
presence of blue- and red-detuned modes that can be used to create an optical
two-color trap.

###  III.1 Design requirements

The design of a periodic waveguide aimed at increasing the coupling with cold
atoms should maximize the emission rate into the waveguide mode
$\Gamma_{1\textrm{D}}$. Since the group index diverges at a band edge, the
first idea is to align the transition frequency of the atom with any band edge
of the photonic dispersion diagram. Unfortunately, this simple design cannot
work for several reasons.

First, slow light is very sensitive to fabrication imperfections, resulting in
backscattering, radiation losses, and potentially light localization
HughesPRL05 ; topolancik2007experimental ; mazoyer2009disorder ; FaggianiSR16
. Therefore, in a practical situation, fabrication imperfections set an upper
bound to the group index $n_{g}$ that can be reached patterson2009disorder ;
garcia2010density ; MazoyerOE10 . To improve the fabrication tolerance, X.
Zhang _et al._ proposed to use parabolic dispersion curves with large
effective photon masses zang2016interaction . Therefore, the design should not
only increase the group index, but also reduce the curvature of the dispersion
relation.

A second issue is the value of the effective mode area at the position of the
atom, or equivalently the effective cross section of the photon seen by the
atom. One should be careful not to lose with an increased mode area what is
gained with an increased group index. Hence, the design has to find a trade-
off between two opposite trends. On the one hand, we need a mode whose field
extends far into the air cladding, implying that it weakly interacts with the
periodic pattern. On the other hand, we need to control the group velocity and
the curvature of the dispersion relation, meaning that we need a mode that
strongly interacts with the periodic pattern.

Finally, a third important challenge is to generate a stable optical trap for
the atoms at subwavelength distances of the waveguide and to be able to bring
the atoms inside the trap. A fully integrated trapping scheme can be achieved
by using “fast” guided modes (i.e., with standard values of $n_{g}$) at
frequencies detuned from the atomic transition. For instance, red- and blue-
detuned modes can be used to create a two-color trap le2004atom . The design
of the periodic waveguide should thus ensure the presence of additional modes
with adequate field profiles, which spatially overlap with each other and with
the slow mode.

As a whole, the design of a periodic waveguide with increased atom-photon
interactions is a complex task that should meet the following criteria:

  * •

a slow and single-mode operation at the transition frequency of the atom
(large $n_{g}$),

  * •

a large fraction of the electric field in air outside the structure (small
$A_{\textrm{eff}}$),

  * •

a large effective photon mass for an improved robustness to fabrication
imperfections,

  * •

the existence of additional modes at frequencies detuned from the atomic
transition for trapping the atoms optically with low powers (few mW),

  * •

a clear access around the structure to ease the transport of the atoms to the
trapping sites.

Up to now, two main geometries have been investigated. The first one, the
alligator waveguide, is composed of a tiny 250-nm-wide air slot symmetrically
surrounded by two corrugated bridge waveguides goban2014atom ;
goban2015superradiance ; yu2014nanowire . This geometry fulfills the first two
criteria (large $n_{g}$ and small $A_{\textrm{eff}}$) as well as the fourth
one. However, it has been shown in zang2016interaction  that the effective
photon mass of the slow mode is small, meaning that alligator waveguides are
very sensitive to fabrication imperfections in the slow-light regime. The
fifth criterion is not met either since the atoms need to be loaded in a
narrow interacting region, which constitutes a major experimental challenge
burgers2019clocked . External trapping via side illumination has been
considered kimble2020optica ; kimble2020pnas , but a stable trapping of atoms
_inside_ the air slot has not been demonstrated. As a result, the value of
$\beta$ and the number $N$ of trapped atoms are limited to $\beta\sim 0.5$ and
$N\sim 3$ goban2015superradiance ; hood2016atom .

A different geometry has been proposed in zang2016interaction  to address
these limitations. It consists of a hybrid-clad waveguide that combines two
guidance mechanisms: total internal reflection on one lateral side with a
sharp sidewall and photonic bandgap on the opposite lateral side with a two-
dimensional photonic crystal. The hybrid-clad waveguide provides a flatter
dispersion curve than the alligator waveguide (50 times enhanced effective
photon mass) and a 2$\pi$ solid-angle access to the interaction region.
However, the possibility to use detuned guided beams for trapping the atoms
(fourth criterion) has not been investigated yet. One the one hand, the
bandgap of the photonic-crystal cladding sets important constraints on the
spectral range available to place additional guided modes that could be used
for trapping. On the other hand, the photonic-crystal geometry provides
various degrees of freedom for engineering the band diagram.

In the following, we design an asymmetric comb waveguide that fulfills all
five criteria. In addition, the dispersion relation of the slow mode is
quartic instead of quadratic. We show that, thanks to the quartic dispersion,
the slow mode of the comb waveguide is more tolerant to fabrication
imperfections than the mode of the hybrid-clad waveguide proposed in
zang2016interaction  that mostly follows a standard parabolic dispersion.

###  III.2 Asymetric comb waveguide

Let us consider a GaInP membrane (refractive index $n=3.31$) with a thickness
$t_{y}=150$ nm, as shown in Fig. 1. Starting from the 2D structure in Fig.
2(d), we have designed a 3D comb waveguide that supports a slow mode with a
quartic dispersion $\omega-\omega_{e}\propto-(k-\pi/a)^{4}$ and a group index
$n_{g}=50$ at $\lambda_{0}=780$ nm, the wavelength of the
$5S_{1/2}\leftrightarrow 5P_{3/2}$ transition of Rb atoms. The geometrical
parameters are $a=283$ nm, $H=2a$, $H_{\textrm{etched}}=0.8H$, and $w=0.422a$.
The band diagram of this comb waveguide is shown in Fig. 3(a); the slow mode
with a quartic dispersion is highlighted in green.

Figures 3(b) and 3(c) display the distribution in the $(x,z)$ plane of the
dominant components of the electric field ($E_{x}$ and $E_{z}$) at
$\lambda_{0}=780$ nm. Note that the evanescent tail of the electric field
extends far in the air cladding. On the side opposite the teeth, the
longitudinal electric-field component $E_{z}$ is maximum in front of the
teeth, at $z\equiv a/2$ (mod $a$). The positions of the maxima of the
transverse electric-field component $E_{x}$ are shifted by $a/2$, at $z\equiv
0$ (mod $a$).

Figure 3: Slow mode with a quartic dispersion. (a) Band diagram of the 3D comb
waveguide designed to support a slow mode with a quartic dispersion (green
curve) that stronly interacts with Rubidium atoms. The geometrical parameters
are $a=283$ nm, $H=2a$, $H_{\textrm{etched}}=0.8H$, $w=0.422a$, $t_{y}=150$ nm
and $n=3.31$. The red and light blue dots mark the “fast” modes red-
($\lambda_{r}=837$ nm) and blue-detuned ($\lambda_{b}=736$ nm) with respect to
the atomic transition that are used for trapping, as detailed in Section IV.
(b) Amplitude of the $x$-component of the electric field of the slow mode. (c)
Amplitude of the $z$-component of the electric field of the slow mode. The
field distributions are calculated at $\lambda_{0}=780$ nm in the $(x,z)$
plane ($y=0$ in the middle of the membrane). (d) Group index as a function of
the distance to the band edge, $\Delta\lambda=|\lambda-\lambda_{e}|$
(logarithmic scales). (e) Effective photon mass as a function of
$\Delta\lambda=|\lambda-\lambda_{e}|$ (logarithmic scales). The green curves
correspond to the slow mode of the 3D comb waveguide [green curve in (a)] and
the orange curves correspond to the hybrid-clad waveguide in
zang2016interaction . In (d), the dashed curves indicate the
$\Delta\lambda^{-3/4}$ (green) and $\Delta\lambda^{-1/2}$ (orange) scaling
laws.

In addition to the slow mode, the band diagram should also provide two
additional modes, red- and blue-detuned with respect to the atomic transition,
that can be used to realize an optical two-color trap. These modes should lie
below the light line and below the frequency of the electronic bandgap of
GaInP ($\lambda=680$ nm). The design of the structure has been performed by
taking into account these additional constraints. The red- and blue-detuned
modes are highlighted by the red and light blue dots in Fig. 3(a). The
calculation of the trapping potential is presented in Section IV.

To ease the future implementation of the structure, it is important to keep in
mind that nanostructured devices are never manufactured to their nominal
specifications. Hence, real periodic waveguides are not exactly periodic and
suffer from random fabrication imperfections. Slow light is particularly
sensitive to these imperfections. It experiences random scattering as it
propagates, resulting in backscattering, radiation losses, and possibly light
localization HughesPRL05 ; topolancik2007experimental ; mazoyer2009disorder .

In the usual case of a parabolic dispersion, it has been shown in
zang2016interaction ; FaggianiSR16  that the important physical parameter to
be considered to evaluate the sensitivity of slow light to fabrication
imperfections is the effective photon mass,
$m_{\textrm{eff}}=\left(\partial^{2}\omega/\partial k^{2}\right)^{-1}$. It is
a constant for a parabolic dispersion. X. Zhang _et al._ concluded that one
should consider a dispersion curve with a large effective photon mass to
improve the fabrication tolerance zang2016interaction .

However, since the dispersion curve of the asymmetric comb waveguide is not
parabolic, we cannot use this unique figure of merit to assess the tolerance
of the slow mode to fabrication imperfections. Therefore, we have extended the
reasoning in zang2016interaction  to a general, non-parabolic, dispersion. In
this case, one must consider two different physical parameters: the variation
of the group velocity with the frequency, $\partial v_{g}/\partial\omega$, and
the distance of the operation frequency to the band edge,
$\Delta\omega=|\omega-\omega_{e}|$. The first parameter $\partial
v_{g}/\partial\omega$ is inversely proportional to the effective photon mass,
which is not a constant in the case of a non-parabolic dispersion. On the
other hand, the second parameter $\Delta\omega=|\omega-\omega_{e}|$ is
independent of $m_{\textrm{eff}}$ if the dispersion is not parabolic. To
improve the fabrication tolerance, one has to increase both parameters: the
effective photon mass _at the operation frequency_ and the distance of the
operation frequency to the band edge.

We have calculated these figures of merit for the asymmetric comb. They are
shown by the green curves in Figs. 3(d) and 3(e). For the sake of comparison,
we have also calculated the same parameters for the hybrid-clad waveguide
proposed in zang2016interaction , which is the most resilient to fabrication
imperfections in the waveguide QED literature. Figure 3(d) displays the
distance of the operation wavelength to the band edge,
$\Delta\lambda=|\lambda-\lambda_{e}|$, as a function of the group index.
Regardless of the value of the group index, the comb waveguide allows one to
operate further from the band edge than the hybrid-clad waveguide. The dashed
green and orange curves show that the asymmetric comb waveguide follows the
scaling law that is expected for a quartic dispersion while the hybrid-clad
waveguide mostly follows the scaling law of a quadratic dispersion. The dashed
horizontal line marks the value $n_{g}=50$ that is used in the following.
Figure 3(e) shows the effective photon mass as a function of $\Delta\lambda$.
Regardless of the operation frequency, the asymmetric comb has a larger
effective mass than the hybrid-clad waveguide. These results allow us to
conclude that the asymmetric comb waveguide is more tolerant to fabrication
imperfections than the hybrid-clad waveguide in zang2016interaction .

##  IV Two-color optical trap for Rubidium atoms

We show hereafter that the versatility of the asymmetric comb allows us to
trap Rb atoms optically as close as 100 nm from the waveguide. At these deeply
subwavelength distances, the atoms can strongly interact with the slow mode
aligned with their transition frequency. The objective is to trap atoms where
the electric field of the slow mode is intense, see Figs. 3(b)-(c). For
accessibility reasons, it is not appropriate to create a trap between the comb
teeth. Therefore, we aim at trapping atoms on the side opposite the teeth, see
the red dot in Fig. 1.

There exist various ways to trap atoms close to a dielectric structure
balykin2004atom . In waveguide QED, a smart, fully integrated, approach
consists in taking advantage of the waveguide to build an optical trap with
guided light beams. In this work, we use a two-color trap, whose principle is
to send two additional light beams into the waveguide le2004atom . The first
one is red detuned with respect to the atomic transition; it produces a
negative Stark shift on the energy of the ground state $5S_{1/2}$ that results
in an attracting potential $U_{r}(x,y,z)<0$. The second beam is blue detuned;
it creates a positive Stark shift and a repulsive potential $U_{b}(x,y,z)>0$.
The combination of the two potentials,

$U_{\textrm{trap}}(x,y,z)=U_{r}(x,y,z)+U_{b}(x,y,z),$ (1)

can produce a deep potential well (i.e., a stable trap) at a given position in
space that depends on the spatial profile of both guided beams, on their
powers, and on the polarizability of the atom.

On top of the trapping potential $U_{\textrm{trap}}$ generated by the detuned
beams, it is necessary to take into account the attracting Casimir Polder (CP)
potential that arises when an atom interacts with electromagnetic vacuum
fluctuations near a dielectric surface casimir1948influence ;
buhmann2007dispersion . We assume that the CP potential felt by an atom close
to the vertical sidewall of the comb (see the red dot in Fig. 1) is the same
as the potential near a planar surface, which varies as

$U_{\textrm{CP}}=-\frac{C_{3}}{d^{3}},$ (2)

where $d$ is the distance between the atom and the surface and $C_{3}$ is a
constant that depends both on the material and the atom. The value of $C_{3}$
for GaInP and Rb atoms, $C_{3}\simeq 6.7\times 10^{4}$ mK.nm3, is computed
with the formula in caride2005dependences ; the permittivity of GaInP is taken
from the experimental data in schubert1995optical . The desired depth of the
trapping potential for cold atoms is typically of the order of 1 mK. The CP
potential is thus dominant close to the surface and it becomes negligible for
distances larger than $100$ nm: $U_{\textrm{CP}}(10~{}\textrm{nm})\sim-67$ mK
and $U_{\textrm{CP}}(100~{}\textrm{nm})\sim-6.7\times 10^{-2}$ mK. The CP
potential makes it very difficult to trap atoms stably at distances well below
100 nm.

Figure 4: Electric-field distributions of the blue- and the red-detuned guided
modes used for the two-color trap. (a)-(b) Blue-detuned mode $\mathbf{E}_{b}$
at $\lambda_{b}=736$ nm, marked by a light blue dot in the band diagram of
Fig. 3(a). (c)-(d) Red-detuned mode $\mathbf{E}_{r}$ at $\lambda_{r}=837$ nm,
marked by a red dot in the band diagram of Fig. 3(a). In (a) and (c) the field
is represented in the $(x,z)$ plane at $y=0$ (cross-section through the center
of the GaInP membrane). The solid lines show the contour of the comb
waveguide. In (b) and (d) the field is represented in a $(y,z)$ plane located
in air at a distance $d=100$ nm from the structure.

In order to realize a two-color trap with guided modes, the frequencies of
both detuned signals have to be chosen with the band diagram in Fig. 3(a). Let
us emphasize that, since the waveguide is periodic, the guided modes are Bloch
modes and their field distributions are periodically modulated along the $z$
direction. It is thus possible to create a periodic array of potential wells
in the $z$ direction without using contra-propagating beams. It is convenient
to choose the red-detuned frequency on the same band as the slow mode (green
band), but closer to the light line where the group velocity is of the order
of $c/n$. Then, one of the higher-order bands can be used for the blue-detuned
frequency.

Since the wavelength of the $5S_{1/2}\leftrightarrow 5P_{3/2}$ transition of
Rb atoms is $\lambda_{0}=780$ nm, we have chosen to work with
$\lambda_{r}=837$ nm for the red-detuned field and $\lambda_{b}=736$ nm for
the blue-detuned field. These wavelengths are displayed in Fig. 3(a) with red
and light blue dots. The electric fields of the corresponding guided modes are
shown in Fig. 4. The amplitude of the red-detuned mode is almost homogeneous
in air along the $z$ direction while the amplitude of the blue-detuned mode
exhibits a more pronounced periodic modulation, with intensity maxima in front
of the comb teeth at $z\equiv a/2$ (mod $a$). With these two fields, an atom
near the back of the comb will “feel” an attracting potential that is almost
independent of $z$ and a repulsive potential that is stronger in front of the
teeth. By playing with the relative powers of both detuned beams, it is thus
possible to create a potential well with a miniumum at $z\equiv 0$ (mod $a$).
At this position, the transverse component of the electric field of the slow
mode is maximum, see Fig. 3(b). Therefore, an atom trapped at this position
will interact efficiently with the slow mode.

Figure 5:  Calculated potential of the two-color trap. (a)-(b) Distribution of
the trapping potential $U_{\textrm{trap}}$ created by the superposition of the
blue- and red-detuned modes. As in Fig. 4, we represent a cross-section in the
$(x,z)$ plane at $y=0$ (a) and a cross-section in a $(y,z)$ plane located in
air at a distance $d=100$ nm from the waveguide (b). Red dots represent atoms
trapped in the potential wells. The gray area in (a) shows the comb waveguide,
where the potential is not defined. The dashed lines in (b) remind the
position of the GaInP membrane that is located in a different plane, shifted
by 100 nm in the $x$ direction. (c) Trapping potential $U_{\textrm{trap}}$
(dashed curves) and total potential $U_{\textrm{trap}}+U_{\textrm{CP}}$ (solid
curves) as a function of the distance $d$ to the waveguide for $y=0$ and $z=0$
(mod $a$). Two different sets of trapping powers are represented. The powers
$(P_{r_{1}},P_{b_{1}})=(1.6,1.3)$ mW create a potential well at $d=100$ nm
with a depth $U\approx-2.2$ mK. The powers $(P_{r_{2}},P_{b_{2}})=(1,1.3)$ mW
create a potential well at $d=160$ nm with a depth $U\approx-0.8$ mK. (d)
Total potential $U_{\textrm{trap}}+U_{\textrm{CP}}$ as a function of the
vertical coordinate $y$ for $z=0$ (mod $a$). The orange (resp. light blue)
curve shows the potential created at a distance $d=100$ nm (resp. $d=160$ nm)
by blue- and red-detuned modes with powers $(P_{r_{1}},P_{b_{1}})$ [resp.
$(P_{r_{2}},P_{b_{2}})$].

To be quantitative, we have computed the trapping potential
$U_{\textrm{trap}}(x,y,z)$ for Rb atoms with the recently-introduced open-
source package Nanotrappy berroir2022nanotrappy . The inputs of Nanotrappy are
the electric fields of the blue- and red-detuned guided modes, which are
calculated with the mode solver for periodic waveguides used to calculate the
band diagram LecampOE07 . Nanotrappy calculates the Stark shifts induced by
these electric fields and the corresponding potentials. The calculation
includes the scalar, vector, and tensor shifts berroir2022nanotrappy .

Figures 5(a)-(b) display maps of the trapping potential $U_{\textrm{trap}}$ in
the $(x,z)$ plane at $y=0$ and in a $(y,z)$ plane located in air at a distance
$d=100$ nm from the structure. The powers of the red- and blue-detuned modes
are respectively $(P_{r_{1}},P_{b_{1}})=(1.6,1.3)$ mW. The potential is given
in mK so that it can be easily compared with the temperature $T$ of cold Rb
atoms in a typical experiment, $T\sim 10~{}\mu$K. Red dots highlight the
potential minima, where atoms can be trapped, at $y=0$, $z=0$ (mod $a=283$
nm), and a distance $d=100$ nm from the structure. The confinement is good in
all three directions of space.

Finally, we investigate in Figs. 5(c)-(d) the impact on the trap of the powers
of the detuned modes, as well as the effect of taking into account the CP
potential. Cross-sections of the trapping potential $U_{\textrm{trap}}$
(dashed curves) and of the total potential $U_{\textrm{trap}}+U_{\textrm{CP}}$
(solid curves) are plotted for two different sets of trapping powers
$(P_{r_{1}},P_{b_{1}})=(1.6,1.3)$ mW and $(P_{r_{2}},P_{b_{2}})=(1,1.3)$ mW.
Note that the power of the blue-detuned mode (i.e., the repulsive potential
$U_{b}$) is kept constant. Decreasing the power of the red-detuned mode has
two main effects. First, the depth of the potential well is reduced from
$\approx 2.2$ mK to $\approx 0.8$ mK. Second, the trapping distance is
increased from $100$ nm to $160$ nm. At such distances, adding the CP
contribution does not modify significantly the potential well. It is only
modified at shorter distances where the attracting CP interactions become
dominant. For the closest trap ($d=100$ nm, orange solid curve), the potential
barrier between the trap and the structure is lowered by the CP interactions
but it remains higher than 1 mK. The orange curve in Fig. 5(d) shows that the
potential minimum is not necessarily located at $y=0$. The height of the tiny
potential barrier at $y=0$ is only 50 $\mu$K

We have demonstrated the possibility to trap cold Rb atoms at deeply
subwavelength distances ($d=100$ nm) from the comb waveguide where the
$x$-component of the electric field of the slow mode is intense. We have used
a two-color optical trap with powers of about 1 mW, a value well compatible
with the powers used in current integrated-optics experiments. We show in the
following Section that an atom trapped at this position interacts strongly
with the slow mode.

##  V Strong atom-photon interaction

Having established the possibility to trap Rb atoms close to the comb
waveguide, we now show that trapped atoms can strongly interact with the slow
mode. For that purpose, we compute the spontaneous emission rate of an excited
atom and the $\beta$ factor – i.e., the fraction of light coupled to the slow
mode.

With the optical trap discussed in Section IV, atoms are located at positions
where the $x$-component of the electric field of the slow mode is dominant.
Therefore, we consider in the following spontaneous emission of an atom with a
dipole $\mathbf{d}$ that is linearly polarized along the $x$ direction,
$\mathbf{d}=d\mathbf{e}_{x}$ with $\mathbf{e}_{x}$ the unitary vector along
the $x$ direction. Experimentally, this situation can be realized by imposing
a quantization axis along $x$ with an external magnetic field.

We calculate the total emission rate $\Gamma_{\textrm{tot}}$ and the emission
rate in the slow mode $\Gamma_{1\textrm{D}}$ (sum of the forward and backward
propagation directions). The former is proportional to the imaginary part of
the total Green tensor at the position $\mathbf{r}_{0}$ of the atom Novotny ,

$\frac{\Gamma_{\textrm{tot}}}{\Gamma_{0}}=\frac{6\pi
c}{\omega_{0}}\mathbf{e}_{x}\cdot\textrm{Im}\left[\mathbf{G}(\mathbf{r}_{0},\mathbf{r}_{0},\omega_{0})\right]\mathbf{e}_{x},$
(3)

while the latter depends only on the electric field of the guided mode
goban2014atom ; lecamp2007prl ,

$\frac{\Gamma_{1\textrm{D}}}{\Gamma_{0}}=\frac{n_{g}\sigma
a\varepsilon_{0}|\mathbf{e}_{x}\cdot\mathbf{E}_{m}^{\ast}(\mathbf{r}_{0})|^{2}}{2\iiint_{\textrm{Cell}}\varepsilon|\mathbf{E}_{m}|^{2}d^{3}\mathbf{r}}\,.$
(4)

In these expressions, $\omega_{0}$ is the transition frequency,
$\Gamma_{0}=\omega_{0}^{3}|\mathbf{d}|^{2}/(3\pi\hbar\varepsilon_{0}c^{3})$ is
the emission rate in vacuum, $\sigma=3\lambda_{0}^{2}/(2\pi)$ is the
absorption cross-section, $\mathbf{E}_{m}$ is the electric field of the slow
mode, and $n_{g}=c/v_{g}$ is its group index. One usually defines the mode
effective area as
$A_{\textrm{eff}}(\mathbf{r}_{0})=\iiint_{\textrm{Cell}}\varepsilon|\mathbf{E}_{m}|^{2}d^{3}\mathbf{r}/(a\varepsilon_{0}|\mathbf{e}_{x}\cdot\mathbf{E}_{m}^{\ast}(\mathbf{r}_{0})|^{2})$.
Note that the volume integral runs over one unit cell of the periodic
waveguide.

Then, we deduce the emission rate outside the slow mode (i.e., into all the
other modes), $\Gamma^{\prime}=\Gamma_{\textrm{tot}}-\Gamma_{1\textrm{D}}$, as
well as the value of the $\beta$ factor,
$\beta=\Gamma_{1\textrm{D}}/\Gamma_{\textrm{tot}}$. Let us recall that the
slow mode supported by the comb waveguide designed in Section III has a group
index $n_{g}=50$ at the transition wavelength of Rb atoms, $\lambda_{0}=2\pi
c/\omega_{0}=780$ nm. Its electric field is represented in Figs. 3(b)-(c).

The main difficulty is the calculation of the total Green tensor
$\mathbf{G}(\mathbf{r}_{0},\mathbf{r}_{0},\omega_{0})$ of an infinitely long
periodic waveguide, which requires an accurate calculation of the emission
into radiation modes with outgoing-wave conditions in a periodic medium
LecampOE07 . Two different approaches are often used to bypass this
difficulty. The first one amounts to assume that the emission rate into
radiation modes is approximately equal to the emission rate in vacuum,
$\Gamma^{\prime}\approx\Gamma_{0}$, see for instance zang2016interaction .
With this approximation, Eq. (3) simply becomes
$\Gamma_{\textrm{tot}}/\Gamma_{0}\approx 1+\Gamma_{1\textrm{D}}/\Gamma_{0}$.
In that case, the calculation only requires the knowledge of the guided mode,
which can be calculated with a Bloch-mode solver. The second approach that
avoids the calculation of the Green tensor of an infinitely long periodic
waveguide consists in considering a finite-size structure formed by a finite
number of periods, see for instance hung2013trapped . The main drawback is
that the calculated structure (a Fabry-Perot cavity) is different from the
desired one (an infinitely long waveguide). As a result, the emission rate
exhibits a series of spurious resonance peaks that depends on the arbitrary
choice of the structure length and termination. It is thus difficult to infer
the actual emission rate of the periodic waveguide.

In contrast to these two approximate approaches, we calculate rigorously the
Green tensor of the periodic waveguide by using a modal method that relies on
an exact Bloch-mode expansion LecampOE07 . Figure 6 shows the variation of the
decay rates $\Gamma_{\textrm{tot}}$, $\Gamma_{\textrm{1D}}$, $\Gamma^{\prime}$
(first line), and of the $\beta$ factor (second line) as a function of the
position of the atom. In Fig. 6(a), the atom is moved horizontally away from
the comb for $y=0$ and $z=0$ (mod $a$). In Fig. 6(b), the atom is moved
vertically for $d=100$ nm and $z=0$ (mod $a$). In Fig. 6(c), the atom is moved
along the waveguide for $d=100$ nm and $y=0$. The yellow areas represent the
volume where atoms are likely to be trapped, defined as the positions where
the value of the orange potential in Fig. 5 is between $U_{min}\approx-2.2$ mK
and $U_{min}+50$ $\mu$K. This defines a trapping volume of typical size 30 nm
along $x$, 160 nm along $y$ and 20 nm along $z$.

For large distances, $d>400$ nm, the emission rate $\Gamma_{\textrm{1D}}$ into
the slow mode is negligible compared to the emission rate $\Gamma^{\prime}$
into the radiation continuum and the $\beta$ factor tends towards zero. As the
distance $d$ decreases, the atom enters the region where the field of the slow
mode is intense and Figs. 6(a) shows a strong enhancement of
$\Gamma_{\textrm{1D}}$ that results in an increase of the $\beta$ factor. For
a trapping distance of $d=100$ nm (yellow area),
$\Gamma_{\textrm{1D}}=10\Gamma_{0}$, $\Gamma^{\prime}=1.3\Gamma_{0}$, and
$\beta=0.88$. This number is significantly larger than the value of
$\beta\approx 0.5$ that has been experimentally observed for the alligator
waveguide goban2015superradiance ; hood2016atom .

In Figs. 6(b)-(c), we fix the distance $d=100$ nm and we vary the position of
the atom in the two other directions. Along the vertical direction, as long as
the atom is located inside the trap in front of the structure
($-75~{}\textrm{nm}\leqslant y\leqslant 75~{}\textrm{nm}$), the decay rate
into the slow mode remains dominant $\Gamma_{\textrm{1D}}>8\Gamma_{0}$ and
$\beta>0.8$. Along the $z$ direction, $\Gamma_{\textrm{1D}}$ varies
periodically. Its variation over one period is directly related to the
variation of $|E_{x}(z)|$ represented in Fig. 3(b). On the other hand,
$\Gamma^{\prime}$ is almost constant.

Figure 6: Decay rates of an atom near the 3D asymmetric comb waveguide. First
line: Decay rates $\Gamma_{\textrm{tot}}$ (blue), $\Gamma_{\textrm{1D}}$
(red), and $\Gamma^{\prime}$ (black) as a function of the distance $d$ from
the waveguide for $y=0$ and $z=0$ (mod $a$) (a), as a function of $y$ for
$d=100$ nm and $z=0$ (mod $a$) (b), and as a function of $z$ for $d=100$ nm
and $y=0$ (c). All decay rates are normalized by the decay rate $\Gamma_{0}$
of the atom in vacuum. Second line: Variation of the $\beta$ factor,
$\beta=\Gamma_{\textrm{1D}}/\Gamma_{\textrm{tot}}$, at the same positions. The
yellow areas represent the volume where atoms are likely to be trapped,
defined as the positions where the value of the orange potential in Fig. 5 is
between $U_{min}\approx-2.2$ mK and $U_{min}+50$ $\mu$K.

Finally, let us emphasize that the presence of the comb increases the emission
rate into the radiation continuum $\Gamma^{\prime}$, compared to the emission
rate in vacuum $\Gamma_{0}$. Indeed, for $d\leqslant 100$ nm and
$-75~{}\textrm{nm}\leqslant y\leqslant 75~{}\textrm{nm}$,
$\Gamma^{\prime}\geqslant 1.3\Gamma_{0}$. Therefore, the assumption
$\Gamma^{\prime}\approx\Gamma_{0}$ would lead to an error larger than $30\%$
for $d\leqslant 100$ nm. This trend of an increase of $\Gamma^{\prime}$ close
to a dielectric structure is not general; a previous calculation with the same
numerical method has shown a decrease of $\Gamma^{\prime}$ in the near field
of a different periodic waveguide LalanneOL2015 .

##  VI Conclusion

We have proposed a new waveguide geometry, the asymmetric comb, that provides
a strong interaction between trapped atoms and guided photons. An important
originality of the structure is its quartic dispersion relation of the form
$\omega-\omega_{e}\propto-(k-\pi/a)^{4}$, unique in the context of waveguide
QED. We have demonstrated that this specific form of the dispersion relation
reduces the impact of the inevitable fabrication imperfections. Then, we have
shown how cold Rubidium atoms can be trapped at subwavelength distances
($d=100$ nm) from the structure by implementing a two-color optical trap with
guided modes that are red and blue detuned with respect to the atomic
transition frequency. Finally, we have completely characterized the decay of
an excited atom in this complex photonic environment by calculating rigorously
the decay rate $\Gamma_{1\textrm{D}}$ into the guided mode as well as the
decay rate $\Gamma^{\prime}$ into all other radiative channels. Atoms inside
the trap decay preferentially into the slow mode (group velocity $v_{g}=c/50$)
with a $\beta$ factor as high as 0.88.

We have conducted preliminary studies (not shown here) that show that the
fabrication of the structure is completely feasible. We have considered
throughout the paper a structure made of GaInP, a semiconductor material with
a high refractive index. This choice is not critical for the design. We have
checked that an asymmetric comb waveguide with a quartic dispersion and a
large $\beta$ factor can also be designed in a material with a lower
refractive index such as silicon nitride or silicon oxide.

Being able to control the emission with figures of merit as large as
$\Gamma_{1\textrm{D}}=10\Gamma_{0}$, $\Gamma^{\prime}=1.3\Gamma_{0}$, and
$\beta=0.88$ puts the strong coupling regime of waveguide QED
($\Gamma_{1\textrm{D}}/\Gamma^{\prime}\gg 1$) within reach. In this important
regime, both collective plankensteiner2015selective ; asenjo2017exponential ;
kornovan2019extremely ; shlesinger2021time ; ferioli2021storage ;
reitz2022cooperative  and non-linear quantum chang2014quantum  phenomena are
enhanced, meaning that single photon switches chang2007single  or coherent
photon storage gouraud2015demonstration ; sayrin2015storage ; solano2017super
could be achieved with high efficiencies. Exotic many-body phenomena such as
correlated photon transport mahmoodian2020dynamics , many-body localization
fayard2021many , or fermionization of the multiple excited states
albrecht2019subradiant  could also be explored with the asymmetric comb.
Finally, let us emphasize that the uncommon dispersion relation of the slow
mode could dramatically alter the nature and the range of the photon-mediated
interactions for atoms whose transition frequencies lie inside or outside the
bandgap gonzalez2015subwavelength ; douglas2015quantum ; zhang2020subradiant .
Enlarging the bandwidth of the structure also allows one to probe with reduced
distortion the fast dynamics of superradiant emission gross1982superradiance .

## Acknowledgements

This work was supported by the French National Research Agency (ANR) under the
NanoStrong Project (ANR-18-CE47-0008) and the Région Île-de-France (DIM
SIRTEQ). This project has also received funding from the European Union’s
Horizon 2020 research and innovation programme under Grant Agreement No.
899275 (DAALI project). A.U. was supported by the European Union (Marie Curie
Fellowship SinglePass 101030421).

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