# Unidirectional absorption, storage, and emission of single photons in a
collectively responding bilayer atomic array

K. E. Ballantine<EMAIL_ADDRESS>J. Ruostekoski
<EMAIL_ADDRESS>Department of Physics, Lancaster University,
Lancaster, LA1 4YB, United Kingdom

###### Abstract

Two-dimensional regular arrays of atoms are a promising platform for quantum
networks, with collective subradiant states providing long-lived storage and
collimated emission allowing for natural coherent links between arrays in free
space. However, a single-layer lattice can only efficiently absorb or emit
light symmetrically in the forward and backward directions. Here we show how a
bilayer lattice can absorb a single photon either incident from a single
direction or an arbitrary superposition of forward and backward propagating
components. The excitation can be stored in a subradiant state, transferred
coherently between different subradiant states, and released, again in an
arbitrary combination of highly collimated forward and backward propagating
components. We explain the directionality of single and bilayer arrays by a
symmetry analysis based on the scattering parities of different multipole
radiation components of collective excitations. The collective modes may
exhibit the conventional half-wave loss of fields near the array interface or
completely eliminate it. The proposed directional control of absorption and
emission paves the way for effective one-dimensional quantum communication
between multiple arrays, with single-photons propagating backward and forward
between quantum information-processing and storage stages.

##  I Introduction

Two-dimensional (2D) atomic arrays interacting with light [1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31] provide a novel quantum-mechanical many-body system to
exploit collective effects in the control of light. Emission from large
lattices is highly collimated [32, 33, 34, 16, 35], leading to efficient
coherent quantum links in free space between them. Quantum networks [36, 37]
require just such coherent links between nodes, while the effective 1D
scattering also eliminates spontaneous emission in different directions, which
is a major loss channel for single atoms or random atomic ensembles [38].
Experimental control of 2D arrays of cold atoms is also rapidly improving [39,
40, 41, 42, 43, 44, 45, 46, 47], and the collective narrowing of the
reflection linewidth beyond the single-atom limit has been recently observed
[47], analogously to the steady-state driving of giant subradiant states of 2D
meta-atom arrays [48]. Further progress towards truly quantum manipulation,
however, requires directional control over the efficient absorption, storage,
and release of propagating single photons.

Subradiant states in free-space 2D arrays of atoms [1, 2, 4, 3, 12, 7, 9, 11,
34, 19, 17, 25, 35] are promising for light storage due to their isolation
from the environment, but, unlike in the case of 1D chains of atoms [49, 50],
in 2D arrays strongly subradiant modes cannot be directly driven by incident
fields. Previous proposals have demonstrated how such modes can be excited in
the steady state, using atomic level shifts to control the orientation of the
dipoles [4, 33], or by optimizing driving [51]. Besides the giant experimental
subradiance of regular subwavelength arrays [47, 48], long-lived states have
also been experimentally observed in disordered atomic clouds [52, 53]. More
recently, schemes to achieve efficient absorption of a time-dependent single-
photon pulse have been put forward [29, 30]. However, for a single thin layer,
complete absorption is only possible when illuminated symmetrically from both
sides. This is because, during the absorption, it is necessary to first excite
the atomic dipoles, which then radiate during the process, leading to
scattering losses. Such scattering can be suppressed by interference with the
incident beam, but only for transmitted light, while reflected light is lost.

Here we show how interference of scattering from a bilayer lattice can be
controlled, depending on symmetry, in order to absorb and emit light in the
forward direction, backward direction, or an arbitrary combination of the two.
We show how a photon incident from a single direction can be absorbed with
probability $0.79$, rising to $0.93$ when the time-dependence of the incident
pulse is optimized, well in excess of the theoretical limit of $0.5$ for a
single layer. The storage occurs by first exciting a superposition of in-plane
excitations, the polarization amplitudes of which are either symmetric or
antisymmetric between the two atomic layers, and then rotating the
polarizations via a linear Zeeman splitting. Each in-plane mode is thus
coupled to a subradiant, out-of-plane mode with corresponding symmetry. We
further demonstrate how coupling between the out-of-plane modes can be used to
control the storage, leading to coherent oscillations between the subradiant
modes, and how, by engineering the correct phase between these modes, the
excitation can be transferred back to the in-plane modes in such a way that it
is emitted in the forward direction, backward direction, or an arbitrary
combination of the two, independent of the incident beam direction. The
proposed scheme for highly directional single photon storage and emission is
also very different from the protocols of single photon sources based on the
atomic ensembles in the Rydberg states [54]. In such a scheme, a single photon
is transferred to an atomic ensemble via Rydberg excitations using, e.g.,
STIRAP, while assuming all the atoms being confined inside the Rydberg dipole
blockade region.

The directionality of light coupling is explained in terms of the odd and even
parity of the scattered light from the collective excitation eigenmodes of the
arrays that depends on their electric and magnetic multipole character. This
is another example of symmetry properties of scattering in planar atomic
arrays that also plays a role in producing Huygens’ surfaces [22] and in
generating a ${\cal PT}$ scattering symmetry [26]. We show that more complex
geometries, with more than two atoms per unit cell, can also be used to
engineer pure magnetic dipole or electric quadrupole modes with the necessary
symmetry.

##  II Light-matter relations

Figure 1: (a) Geometry of atomic bilayer lattice in $yz$ plane with lattice
constant $d$ and layer separation $a$, capable of absorbing and emitting
unidirectional light. (b) Collective excitation eigenmode level shifts
$\delta_{n}$ of the mode with dipoles oscillating in-plane and symmetric
between top and bottom layers $\mathcal{P}_{ys}$, the in-plane antisymmetric
mode $\mathcal{P}_{ya}$, out-of-plane symmetric mode $\mathcal{P}_{xs}$, and
out-of-plane antisymmetric mode $\mathcal{P}_{xa}$. Level shifts of
$\mathcal{P}_{ya}$ and $\mathcal{P}_{ys}$ intersect at $d=0.91\lambda$.
$20\times 20\times 2$ lattice with $a=0.25\lambda$.

We consider a square lattice in the $yz$ plane with lattice constant $d$.
While in Sec. III we briefly recap the case of a single layer, with one atom
per unit cell, and in Sec. V we consider a four-atom unit cell, the majority
of this work focuses on the case of two atoms per unit cell forming a simple
bilayer lattice, each atom of the unit cell located in the different layer, as
shown in Fig. 1. The layer separation is $a$, i.e. atomic positions $x=\pm
a/2$, $y=md$, $z=nd$ for integer $m,n$, as could be achieved by an additional
double-well structure of the trap in the $x$ direction [55, 56] or using
optical tweezers [43]. There is one atom per site and a $J=0\rightarrow
J^{\prime}=1$ transition. Systems with flexible parameter values could be
achieved, e.g., in Sr atom arrays [57]. A single-photon pulse is incident in
the $\pm\hat{\mathbf{{x}}}$ direction with $y$ polarization. Since the
quantization axis is perpendicular to the propagation axis and the
polarization, the incident field drives a combination of the transitions to
the $J^{\prime}=1,m=\pm 1$ levels, while the $m=0$ level remains unoccupied.

Here we consider the interaction of the lattice with a single photon, and so
limit the atomic dynamics to at most one excitation111The analogies and
differences between the dynamics of single-excitation and that of the coupled-
dipole model of classical harmonic oscillators in the limit of low light
intensity are discussed in detail in Ref. [29].. The basic formalism is based
on nonrelativistic electrodynamics [58, 59]. The reduced many-body density
matrix for the atoms can thus be split into one-excitation and zero-excitation
parts,
$\rho=\left|\Psi\right\rangle\left\langle\Psi\right|+p_{G}\left|G\right\rangle\left\langle
G\right|$, with $p_{G}$ the probability that the lattice is in the ground
state $\left|G\right\rangle=\prod_{j}\left|g_{j}\right\rangle$. The single
excitation sector is described by the state $\left|\Psi\right\rangle=\sum_{\mu
j}\mathcal{P}_{\mu}^{(j)}\hat{\sigma}_{\mu j}^{+}\left|G\right\rangle$ with
$\hat{\sigma}_{\mu j}^{+}=\left|e_{\mu j}\right\rangle\left\langle
g_{j}\right|$ the raising operator to level $\mu$ on atom $j$ [35]. (Here, and
in the following, the atomic amplitudes and the light fields refer to the
slowly varying positive-frequency components, with rapid fluctuations
$\sim\exp{(-i\Omega t)}$, at the frequency $\Omega=ck$ of the incident
electric field, filtered out.) The dynamics can be written in matrix form,
with $\mathbf{b}$ containing the amplitude components
$\mathcal{P}_{\mu}^{(j)}$ for $j=1,\ldots,N$ and $\mu=0,\pm 1$ which evolve
according to [35, 60]

$\dot{\mathbf{b}}=i(\mathcal{H}+\delta\mathcal{H})\mathbf{b}+\bm{\zeta}.$ (1)

The $3N\times 3N$ non-Hermitian matrix $\mathcal{H}$ describes interactions
between atoms, with off diagonal components

$i\xi\hat{\mathbf{{e}}}_{\mu}^{\ast}\cdot\mathsf{G}(\mathbf{r}_{j}-\mathbf{r}_{k})\hat{\mathbf{{e}}}_{\nu},$
(2)

where $\xi=6\pi\gamma/k^{3}$ with
$\gamma=\mathcal{D}^{2}k^{3}/(6\pi\hbar\epsilon_{0})$ the single-atom
linewidth and $\mathcal{D}$ the reduced dipole matrix element, and
$\mathsf{G}$ is the standard dipole radiation kernel such that the scattered
field $\epsilon_{0}\mathbf{E}(\mathbf{r})=\mathsf{G}(\mathbf{r})\mathbf{d}$ at
a point $\mathbf{r}$ from a dipole $\mathbf{d}$ at the origin is

$\mathsf{G}(\mathbf{r})\mathbf{d}=-\frac{\mathbf{d}\delta(\mathbf{r})}{3}+\frac{k^{3}e^{ikr}}{4\pi}\left\\{(\hat{\mathbf{{r}}}\times\hat{\mathbf{{d}}})\times\hat{\mathbf{{r}}}\frac{1}{kr}\right.\\\
\left.-\left[3\hat{\mathbf{{r}}}(\hat{\mathbf{{r}}}\cdot\hat{\mathbf{{d}}})-\hat{\mathbf{{d}}}\right]\left[\frac{i}{(kr)^{2}}-\frac{1}{(kr)^{3}}\right]\right\\},$
(3)

with $r=|\mathbf{r}|$ and $\hat{r}=\mathbf{r}/r$ [61]. Note the contact term
is inconsequential to the physics for ideal dipoles [62, 60], and vanishes for
hard-core bosons which cannot overlap. The diagonal entries of $\mathcal{H}$
are $\delta\Omega+i\gamma$, where $\delta\Omega=\Omega-\omega$ is the detuning
of the incident field from the single-atom resonance $\omega$. The diagonal
$\delta{\cal H}$ contains additional level shifts
$\Delta_{\mu}^{(j)}=\omega-\omega^{(j)}_{\mu}$. The recoil of the atoms [63]
is assumed to be negligible. The single-photon pulse is approximated by a
time-dependent driving term $\bm{\zeta}$, with components

$i(\xi/\mathcal{D})\hat{\mathbf{{e}}}_{\mu}^{\ast}\cdot\epsilon_{0}\bm{\mathbf{\cal
E}}(\mathbf{r}_{j},t)$ (4)

driving level $\mu$ of atom $j$. The incident field is

$\bm{\mathbf{\cal
E}}(\mathbf{r},t)=\mathcal{N}f(x)g(\mathbf{r}_{j})h(t)\hat{\mathbf{{e}}}_{y},$
(5)

with $g(\mathbf{r})$ the Gaussian spatial amplitude, $h(t)$ a time-dependent
envelope which varies on timescales much slower than that set by the optical
frequency, $\mathcal{N}$ a normalization factor chosen such that the total
energy of the beam is $\hbar\Omega$, and an overall propagation phase

$\displaystyle f(x)$ $\displaystyle=\alpha\exp{(ikx)}+\beta\exp{(-ikx)},$ (6)
$\displaystyle=(\alpha+\beta)\cos{(kx)}+i(\alpha-\beta)\sin{(kx)},$ (7)

determining the incident direction and symmetry, such that
$|\alpha|^{2}+|\beta|^{2}=1$. Hence, $\alpha$ is the amplitude of light
propagating in the $+\hat{\mathbf{{x}}}$ forward direction over time, and
$\beta$ that in the $-\hat{\mathbf{{x}}}$ back direction222We can choose the
lattice to be centered at $x=0$ without loss of generality, since a shift in
the lattice is equivalent to a change in phase of $\alpha$ and $\beta$.. The
analysis of a time-dependent pulse is a key ingredient in understanding the
response to a finite energy single-photon drive. While the classical response
of the lattice to coherent light in the limit of low light intensity would
obey similar dynamics, the resulting state would be very different, with only
the single-photon excitation case resulting in quantum entanglement [29].

##  III Absorption in single layer

For a 2D layer confined to a single plane, the maximum efficiency of
absorption of a pulse of light traveling in a single direction is $0.5$.
Taking $\alpha=1$, for example, gives
$f(x)=\exp{(ikx)}=\cos{(kx)}+i\sin{(kx)}$ in Eq. (6). Then, for a lattice at
$x=0$, only the symmetric $\cos{(kx)}$ component drives the atomic dipoles,
while the remaining part of the incident field with 50% of intensity is
irrelevant. Indeed, the electric field at $x=0$ is exactly the same as for a
standing wave, $\alpha=\beta=1/2$, but this configuration has only half the
total incident energy as the unidirectional beam.

This same limitation can be understood in a different light by considering
that any absorption event, including any possible temporal or spatial
variation of level shifts, has a time-reversed equivalent which corresponds to
an allowed process of the excited lattice emitting the light. For a single
layer, regardless of the state the excitation occupies or the level shifts,
there is no way to break the symmetry between forward and backward, and so all
emission processes have equal total power radiated in both these directions.
Hence, the reverse absorption process will result in scattering losses when
the lattice is not illuminated symmetrically. When light is incident
symmetrically from both sides, it can indeed be absorbed with near-unity
efficiency [64] which can be utilized in storing a single photon [29, 30]. For
multiple layers, as described further in Sec. IV.1, this symmetry may be
broken as, e.g., the excitation can be localized on the top or bottom layer,
or there can be different level shifts on each layer.

##  IV Bilayer array

###  IV.1 Collective modes

Although in our numerical analysis we solve the full system of Eqs. (1), the
response of the lattice can be understood by considering the collective
eigenmodes $\mathbf{v}_{n}$ of $\mathcal{H}$, and their eigenvalues
$\delta_{n}+i\upsilon_{n}$, where $\delta_{n}$ is the collective level shift
and $\upsilon_{n}$ is the collective linewidth [65, 66, 1, 67, 68, 60]. For
the applications that we consider in this paper, the bilayer lattice has four
collective eigenmodes of interest. For the $y$-polarized incident field, there
is a symmetric, $y$-polarized mode $\mathbf{v}_{ys}$ with amplitude
$\mathcal{P}_{ys}$, where $\mathcal{P}_{n}=\mathbf{v}_{n}^{T}b$ (since
$\mathcal{H}$ is symmetric
$\mathbf{v}_{j}^{T}\mathbf{v}_{k}=\delta_{jk}$333For some modes,
$\mathbf{v}_{j}^{T}\mathbf{v}_{j}=0$ is also possible, but these are not
encountered in the current system.). This mode consists of all dipoles
oscillating uniformly and in-phase in the $y$ direction (in the plane of the
lattice). There is then also an antisymmetric mode, with amplitude
$\mathcal{P}_{ya}$, where all dipoles within a layer oscillate in phase, but a
$\pi$ phase difference exists between the top and bottom layers, i.e.
$\mathcal{P}_{y}(x=a/2)=-\mathcal{P}_{y}(x=-a/2)$. These modes are coupled to
the out-of-plane modes, one symmetric with amplitude $\mathcal{P}_{xs}$, with
uniform distribution of dipoles in the $x$ direction, and an antisymmetric one
with amplitude $\mathcal{P}_{xa}$, with a $\pi$ phase difference between the
two layers. For a subwavelength lattice there is only the zero-order Bragg
peak for scattering which becomes infinitely sharp in the limit of infinite
lattice size [69, 4, 4, 16, 35]. Because no emission occurs along the dipole
axis, the out-of-plane modes cannot scatter in this direction, and so are
strongly subradiant, with
$\lim_{N\rightarrow\infty}\upsilon_{xs},\upsilon_{xa}=0$ for infinite atom
number $N$.

Figure 2: Interference of light in forward and backward directions due to
differing mode symmetry, for $a=\lambda/4$. (a) Mode amplitude and scattered
field amplitudes in complex plane. If $\mathcal{P}_{s}$ is $\pi/2$ out of
phase with $\mathcal{P}_{a}$, then the scattered fields (shown here up to an
irrelevant global phase factor) interfere constructively, leading to the total
forward scattered field
$\mathbf{E}^{(f)}=\mathbf{E}_{s}^{(f)}+\mathbf{E}_{a}^{(f)}$. If the phase of
$\mathcal{P}_{a}$ varies, while keeping $\mathcal{P}_{s}$ fixed, then
$\mathbf{E}_{a}^{(f)}$ will follow the blue circle, and $\mathbf{E}^{(f)}$ the
black circle. (b) Conversely, $\mathbf{E}_{a}^{(b)}=-\mathbf{E}_{a}^{(f)}$,
and so destructive interference occurs for the relative phase
$\mathcal{P}_{a}=i\mathcal{P}_{s}$, with
$\mathbf{E}^{(b)}=\mathbf{E}_{s}^{(b)}+\mathbf{E}_{a}^{(b)}=0$. As the
relative phase varies and the forward scattering decreases the backward
scattering correspondingly increases.

It is the differing symmetry of these modes which allows full control of the
directionality of both absorption and emission. As described in Sec. III, this
is prevented in the single-layer case by the symmetry between the forward and
backward directions. Writing the total field $\mathbf{E}_{\rm
tot}^{(f,b)}=\bm{\mathbf{\cal E}}^{(f,b)}+\mathbf{E}^{(f,b)}$ propagating in
the forward ($f$) or backward ($b$) direction as the sum of the incident and
scattered field in the same directions, the scattered field can further be
decomposed into
$\mathbf{E}^{(f,b)}=\mathbf{E}_{s}^{(f,b)}+\mathbf{E}_{a}^{(f,b)}$, with
$\mathbf{E}_{s}^{(f,b)}$ ($\mathbf{E}_{a}^{(f,b)}$) the contribution from the
symmetric (antisymmetric) mode. For the bilayer lattice, the symmetric mode
scatters equally in both directions, with even parity,

$\mathbf{E}_{s}^{(f)}=\mathbf{E}_{s}^{(b)},$ (8)

but the antisymmetric mode exhibits odd parity and the scattered field in the
forward direction is the negative of that in the backward direction,

$\mathbf{E}_{a}^{(f)}=-\mathbf{E}_{a}^{(b)}.$ (9)

Hence, when both modes are excited, the interference between the different
contributions in both directions will lead to different total field
amplitudes, breaking the symmetry and allowing for both emission and
absorption even of unidirectional-propagating light. The antisymmetric mode
does not generate the conventional half-wave loss of the field near the atom
array interface that could find useful applications in producing strong near
fields.

For incident light in the forward direction, with $\alpha=1$ in Eq. (6), the
symmetric mode will be driven with an amplitude $\cos{(ka/2)}$ and the
antisymmetric mode with an amplitude $i\sin(ka/2)$. While high-efficiency
absorption can be achieved for arbitrary $a$, this is most simply done for
$a=(2n+1)\lambda/4$, where both symmetric and antisymmetric modes can be
equally excited without extraneous level shifts. We first describe the concept
of suppressed scattering during absorption with $a=\lambda/4$.

The relative phase of scattering is illustrated in Fig. 2. The arrows show the
case of unidirectional absorption or emission, where
$\mathcal{P}_{a}=i\mathcal{P}_{s}$, such that
$\mathbf{E}_{s}^{(f)}=\mathbf{E}_{a}^{(f)}$ and
$\mathbf{E}_{s}^{(b)}=-\mathbf{E}_{a}^{(b)}$. A different phase of
$\mathcal{P}_{a}$ relative to $\mathcal{P}_{s}$ will lead to a different phase
of $\mathbf{E}_{a}^{(f,b)}$ relative to $\mathbf{E}_{s}^{(f,b)}$ and the
interference will, in general, be only partial, with
$\mathcal{P}_{a}=-i\mathcal{P}_{s}$ corresponding to the opposite case of full
destructive interference in the forward direction and constructive in the
backward. This description in terms of symmetric and antisymmetric modes is
also equivalent to a description of interference between scattering from the
top and bottom layers.

The bilayer can also be considered as a regular lattice of two-atom unit
cells, each consisting of a pair of atoms at $\pm a/2\hat{\mathbf{{x}}}$. The
multipole composition of each such unit cell can be found by expanding the
scattered field,

$\mathbf{E}=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\left(\alpha_{E,lm}\bm{\Psi}_{lm}+\alpha_{B,lm}\bm{\Phi}_{lm}\right),$
(10)

where $\bm{\Psi}_{lm}$, $\bm{\Phi}_{lm}$ are the vector spherical harmonics
and $\alpha_{E,lm}$ ($\alpha_{B,lm}$) are electric (magnetic) dipole,
quadrupole, etc., coefficients for $l=1,2,\ldots$, respectively [61, 70]. Then
each of the four collective eigenmodes can be seen as a simple repetition of
one of the modes of the single unit cell. $\mathcal{P}_{ys}$ corresponds to a
uniform repetition of two equal in-phase dipoles that generate a strong
electric dipole, and $\mathcal{P}_{ya}$ to a uniform repetition of two dipoles
of equal magnitude but a $\pi$ phase difference that produces magnetic dipole
and electric quadrupole contributions. More complex non-Bravais lattices, for
example with square or diamond unit cells, can be used to further refine and
control these contributions [22, 21], as discussed in Sec. V. In general, the
multipole scattering can be separated into symmetric (electric dipole,
magnetic quadrupole, etc.) and antisymmetric (magnetic dipole, electric
quadrupole, etc.), when only the overall symmetry is required to engineer the
desired directional constructive or destructive interference [71].

While the in-plane modes couple strongly to the incident $y$-polarized field,
the out-of-plane subradiant modes consist of atomic dipoles oscillating
perpendicular to the light polarization direction. We will generalize the
techniques of Refs. [4, 33] to a multimode case to couple different spatially
delocalized collective modes to each other using an induced level shift for
the atoms. The overall response of the lattice can then be understood in terms
of a simple four-mode-model (see Appendix A). A differential level shift
between the two layers, i.e. a different shift of the $J=0\rightarrow
J^{\prime}=1$ spacing on each layer while keeping the $J^{\prime}=1,m=\pm 1$
levels degenerate, is encapsulated by the effective parameter $\Delta_{a,s}$
in Eq. (21). This differential level shift breaks the symmetry between the
layers and couples each symmetric mode to the corresponding antisymmetric
mode. An overall linear Zeeman splitting of the $m=\pm 1$ levels with
identical values on each layer, meanwhile, is encapsulated by the parameter
$\Delta_{x,y}$ in Eq. (20). This Zeeman splitting breaks the isotropy of the
transition, causing the dipoles to rotate and thereby coupling the in-plane
and out-of-plane modes. The required differential level shifts and Zeeman
splittings could both be achieved by ac Stark shifts of suitably polarized
lasers or microwaves [72], allowing for much faster switching than possible
with magnetic fields.

###  IV.2 Photon absorption

Figure 3: Suppression of transmission while photon absorption is underway. (a)
Spatial distribution of real part ${\rm Re}({\cal E})$, propagating from left
to right. (b) Real part of scattered field ${\rm Re}(E_{y})$. Destructive
interference occurs in backward direction leading to scattering only in
forward direction, which is $\pi$ out-of-phase with incident light. (c) Total
intensity $|\mathbf{E}_{\rm tot}|^{2}$ at $t=0$. Incident light from the left
is absorbed by the array, while for $x>0$ incident field seen in (a) is
canceled by scattered field in (b). $20\times 20\times 2$ lattice illustrated
by black dots, $d=0.91\lambda$, $a=0.25\lambda$.  Figure 4: Absorption of
single photon incident on bilayer from one direction. (a) Total incident power
($P_{\rm in}$) and total transmitted power pulse which is Gaussian in space
($w_{0}=5.8\lambda$) and time ($\gamma\tau=2.3$), when Zeeman splitting is not
present ($P_{0}$), and when coupling $\Delta_{x,y}=10\gamma$ is turned on
briefly at $\gamma t=1.6$ ($P_{\Delta}$), in units of $\gamma\hbar\Omega$. (b)
Corresponding energy in units of $\hbar\Omega$ of all modes in absence of
Zeeman splitting ($E_{0}$), in-plane modes with Zeeman splitting
($E_{y,\Delta}$), and out-of-plane modes with Zeeman splitting
($E_{x,\Delta}$). The coupling is turned off after a period $0.15/\gamma$ to
store excitation. (c) Power and (d) stored energy for pulse which is
exponential in time, $\tau=0.3\gamma$, with and without
$\Delta_{x,y}=10\gamma$ from $t=0$ to $t=0.15\gamma$. $20\times 20\times 2$
lattice with $d=0.91\lambda$, $a=0.25\lambda$.

We consider absorption of a time-dependent forward-propagating single-photon
pulse, with Gaussian time profile $h(t)=\exp{(-t^{2}/\tau^{2})}$ in Eq. (5)
and $\alpha=1$ in Eq. (6). The incident $y$ polarized field will drive a
combination of the in-plane modes $\mathcal{P}_{ys}$ and $\mathcal{P}_{ya}$.
We first treat the case $a=\lambda/4$, where both in-plane modes experience
equal driving from the incident field (see Appendix A), leading to equal
excitation. For this value of $a$, we find the collective line shifts, shown
in Fig. 1, intersect at $d\simeq 0.91\lambda$. Although we will show later how
the photon absorption process is not very sensitive to the values of $d$, this
condition maximizes efficiency by ensuring that each mode oscillates at the
same resonance frequency, therefore keeping the relative phase constant during
the absorption process. For $d\simeq 0.91\lambda$, the antisymmetric in-plane
mode $\mathcal{P}_{ya}$ has a normalized magnetic dipole contribution
$|\alpha_{B,10}|^{2}\simeq 0.63$ from each unit cell and an electric
quadrupole contribution $\sum_{m}|\alpha_{E,2m}|^{2}\simeq 0.37$.

Exciting the atomic array leads to scattering as the process is ongoing. The
spatial distribution of the intensities of the incident, scattered, and total
fields are shown in Fig. 3, for peak incident field amplitude. Scattering from
$\mathcal{P}_{ya}$ and $\mathcal{P}_{ys}$, shown in Fig. 3(b) interfere
destructively in the backward direction leading to zero reflection. While they
interfere constructively in the forward direction, they are out of phase with
the incident light, meaning transmission is also suppressed, as seen in Fig.
3(c). The time dynamics of the process is shown in Fig. 4. In the absence of
shifts between the $m=\pm 1$ levels, the scattering would continue as the
incident pulse fell away, and the pulse would simply radiate forward at a
delayed time, as shown in Fig. 4(a). We will suppress this scattering by
coupling collective modes to each other [4]. At the point of maximum mode
occupation a linear Zeeman splitting $\Delta_{x,y}$ that breaks the degeneracy
between the $m=\pm 1$ levels can be applied uniformly to all atoms. This
causes the polarization to rotate out of the plane, coupling
$\mathcal{P}_{ya}$ to $\mathcal{P}_{xa}$, and $\mathcal{P}_{ys}$ to
$\mathcal{P}_{xs}$, respectively (see Appendix A), as the uniform coupling
preserves the symmetry between layers. The level shifts are then turned off,
and the excitation is stored in the subradiant modes, with
$\upsilon_{xa}\approx 0.001\gamma$ and $\upsilon_{xs}\approx 0.03\gamma$ for
the parameters considered here. The result is shown in Fig. 4(a,b), with
photon absorption efficiency 0.79.

Although the storage mechanism is valid for general lattice separations, the
process is simplest to describe for the case of large spacing between the two
layers in terms of 1D electrodynamics [73], as is done in Appendix A. When the
separation $\lambda\lesssim a\ll\sqrt{A}$, where $A$ is the area of each
layer, propagation between layers is effectively 1D [74, 75, 33, 16]. Then,
each layer is equivalent to a single “superatom” with modified collective
linewidth $\upsilon_{\rm 1D}=3\pi\gamma/(k^{2}d^{2})$ of the uniformly excited
array [69], coupled by a 1D channel [76, 73]. Subradiant states have similarly
been shown to be accessible in a 1D system of waveguide-coupled atoms by
varying the detuning of each atom [73].

The storage efficiency can be improved by understanding the time-reversal
invariance. If the in-plane modes were allowed to decay, they would do so
exponentially, and the resulting intensity would also have an exponential
profile, differing significantly from the Gaussian input. In order to better
match the time-reversed process, we consider also an exponential pulse,
$h(t)=\exp{(t/\tau)}\Theta(-t)$, where $\Theta(t)$ is the Heaviside function.
The result is shown in Fig. 4(c,d), with a higher efficiency of 0.93.

While $d\simeq 0.91\lambda$ results in optimal absorption for $a=0.25\lambda$
with equal collective mode resonances, small variations in $d$ lead only to a
small reduction in the efficiency. For example, we find for $d=0.8\lambda$ an
efficiency $\approx 0.7$ for Gaussian input, rising to $\approx 0.79$ for an
exponential input pulse. The efficiency could be further improved by varying
the level shifts in time during the absorption [30]. However, the differing
collective line shifts, $\delta_{xs}\neq\delta_{xa}$, mean that each mode
evolves at a different frequency, complicating the effect of varying the level
shifts over longer timescales.

High-efficiency absorption is also possible for arbitrary
$a\neq(2n+1)\lambda/4$. To achieve this, we introduce a difference in the
level shifts between the layers at $x=\pm a/2$, as could be achieved by the ac
Stark shift of light with an intensity gradient in the $x$ direction. Upon
introducing different level shifts between each layer, the modes
$\mathcal{P}_{ys}$ and $\mathcal{P}_{ya}$ of $\mathcal{H}$ are no longer
eigenmodes of $\mathcal{H}+\delta{H}$, but are coupled together with nonzero
$\Delta_{a,s}$ in Eq. (21) (see Appendix A). For general $a$,
$\upsilon_{a}\neq\upsilon_{s}$, and to suppress backwards scattering, with
$\mathbf{E}_{s}^{(b)}=-\mathbf{E}_{a}^{(b)}$, we excite a linear combination
with unequal amplitudes $|\mathcal{P}_{s}|\neq|\mathcal{P}_{a}|$. The desired
combination can be targeted by choosing $\Delta_{a,s}$ appropriately. Again,
turning on a linear Zeeman splitting $\Delta_{x,y}$, that breaks the
degeneracy between the $m=\pm 1$ levels at the point of maximum occupation,
transfers the excitation to a corresponding linear combination of
$\mathcal{P}_{xs}$ and $\mathcal{P}_{xa}$, where light can be stored. We
briefly consider a numerical example with $a=0.9\lambda$. For this separation
there is no choice of $d$ such that $\delta_{ys}=\delta_{ya}$. Despite this,
we still find a high efficiency $\approx 0.7$ can be achieved for a Gaussian
input pulse, rising to $\approx 0.85$ when the incident pulse is tailored to
match the time-reversal of emitted light, both for $d=0.9\lambda$.

###  IV.3 Photon retrieval

Figure 5: Retrieval of the photon from storage. (a) Power $P_{f}$ ($P_{b}$) in
the forward (backward) direction, in units $\hbar\gamma\Omega$, when the
photon is released by turning on level shifts $\Delta_{x,y}=10\gamma$ from
$T_{r}=0$ until $\gamma t=0.15$. The excitation is transferred to a
superposition of the in-plane modes where it decays exponentially. $97\%$ of
the total energy is radiated in the forward direction. (b) Total integrated
energy $E_{f}$ ($E_{b}$) in units of $\hbar\Omega$ radiated in the forward
(backward) direction, when the excitation is released at time $T_{r}$. Lattice
parameters as in Fig. 4.

The photon can be released after some time by re-activating the level shifts,
coupling $\mathcal{P}_{xa}$ to $\mathcal{P}_{ya}$, and $\mathcal{P}_{xs}$ to
$\mathcal{P}_{ys}$, respectively, and allowing the excitation to radiate. As
noted in Sec. IV.2, if the photon is stored in a superposition of
$\mathcal{P}_{xs}$ and $\mathcal{P}_{xa}$, then each mode will evolve with a
different frequency, $\delta_{xs}$ and $\delta_{xa}$, respectively. Hence, the
phase difference between the two modes will vary depending on the time,
$T_{r}$, at which the photon is released. This can be used to control the
direction of emission, allowing the photon to be released in the forward
direction, backward direction, or an arbitrary fraction in each, as the
interference depends on the relative phase. The resulting total integrated
power transmitted in each direction, when the level shifts $\Delta_{x,y}$ are
turned on at a time $T_{r}$, is shown in Fig. 5, for the case $a=0.25\lambda$
and $\Delta_{a,s}=0$.

###  IV.4 Storage and coherent oscillations

Figure 6: Coherent oscillations between two subradiant modes. (a) Occupation
of the symmetric and antisymmetric out-of-plane modes when they are coupled by
$\Delta_{a,s}=10\gamma$. (b) Maximum occupation of the symmetric mode $L_{xs}$
(left axis), and frequency of oscillation $\Omega_{\rm osc}$ (right axis), for
different coupling strengths $\Delta_{a,s}$. Solid lines in (b) are derived
from four-mode model. Data in (a) and markers in (b) correspond to numerical
simulation for lattice parameters as in Fig. 4.

As the emission is the time-reversal of absorption, the converse is also true;
an incident photon in an arbitrary combination of left and right incidence can
be absorbed, leading to a specific amplitude and phase relationship of
$\mathcal{P}_{xa}$ and $\mathcal{P}_{xs}$. This storage can be further
controlled by coupling the two out-of-plane modes. A differential level shift
between the layers again gives nonzero $\Delta_{a,s}$ in Eq. (17). This
couples $\mathcal{P}_{xs}$ to $\mathcal{P}_{xa}$, leading to coherent
oscillations between the two populations. The resulting mode occupations can
be defined as [4]

$L_{j}=\frac{|\mathbf{v}_{j}^{T}\mathbf{b}|^{2}}{\sum_{k}|\mathbf{v}_{k}^{T}\mathbf{b}|^{2}},$
(11)

where $j$ indexes all collective modes $\mathbf{v}_{j}$ of ${\cal H}$, as
described in Sec. IV.1. The occupations of the two out-of-plane modes are
shown in Fig. 6(a), with $L_{j}\approx 0$ for all other modes. Coupling
$\mathcal{P}_{x,a}$ and $\mathcal{P}_{x,s}$ thus allows for the occupation to
be redistributed in either a single mode or an equal combination. An
analytical solution of Eq. (17) shows that the oscillations have a frequency
$\Omega_{\rm osc}=2\sqrt{\Delta_{a,s}^{2}+(\delta_{xs}-\delta_{xa})^{2}}$,
while the maximum occupation of, e.g., $\mathcal{P}_{xs}$ is
$1/2+\Delta_{a,s}/\Omega_{\rm osc}$, when starting from a symmetric
superposition of both modes, as illustrated by the solid lines in Fig. 6(b).

##  V Square unit cells

Figure 7: Absorption of a Gaussian pulse, with $\gamma\tau=2.3$, by a square-
unit-cell bilayer. (a) Incident power ($P_{\rm i}$), transmitted power
($P_{0}$) and reflected power ($P_{r}$) in units of $\gamma\hbar\Omega$. (b)
Energy stored in the antisymmetric electric quadrupole mode ($E_{q}$), and the
symmetric electric dipole mode ($E_{d}$), in units of $\hbar\Omega$. Here the
absorption and emission proceed without level shifts. The point of maximum
occupation is marked by the dashed line in (a). $19\times 19\times 4$ lattice
with $d=0.83\lambda$, $a=0.25\lambda$.

In order to show how the symmetry principles of the scattering are more
general, we consider the more complex square-unit-cell bilayer lattice. Here
each unit cell contains four atoms in a square of side-length $a$, oriented in
the $xy$ plane, with the 2D lattice of unit cells with lattice constant $d$
extended in the $yz$ plane. The collective modes of the lattice can be
understood as combinations of the modes of individual unit cells, which can
themselves be decomposed into various electric dipole, magnetic dipole,
electric quadrupole, etc., contributions [29, 22].

Again, the modes fall into a symmetric family (electric dipole, magnetic
quadrupole, etc.), and an antisymmetric family (magnetic dipole, electric
quadrupole, etc.). Exciting one mode from each family can suppress
transmission while the pulse is absorbed, but to achieve a high-fidelity
storage and retrieval requires overlapping mode resonances. We find a
symmetric and antisymmetric collective mode with similar line shifts for
$d\simeq 0.83\lambda$, $a\simeq 0.25\lambda$, for which the electric dipole
and quadrupole modes have $\delta_{n}+i\upsilon_{n}\simeq 1.6+0.66i$ and
$1.6+0.59i$, respectively. These modes exhibit the same symmetry in scattering
as the symmetric and antisymmetric modes, $\mathcal{P}_{ys}$ and
$\mathcal{P}_{ya}$, of the simple bilayer lattice, with
$\mathbf{E}_{d}^{(f)}=\mathbf{E}_{d}^{(b)}$ and
$\mathbf{E}_{q}^{(f)}=-\mathbf{E}_{q}^{(b)}$, where $\mathbf{E}_{d}^{(f,b)}$
and $\mathbf{E}_{q}^{(f,b)}$ are the electric dipole and electric quadrupole
scattered fields, respectively, in the $\pm\hat{\mathbf{{x}}}$ directions. As
in Sec. IV.1, this symmetry allows destructive interference in the reflected
light, and constructive interference in the forward scattered light
suppressing transmission. The principle is the same as demonstrated in Figs. 2
and 3.

The resulting absorption of a uni-directional Gaussian pulse
$h(t)=\exp(-t^{2}/\tau^{2})$ with $\tau=1.1\gamma$ is shown in Fig. 7. In this
case, the pulse is absorbed and re-emitted in the absence of any externally
applied level shifts. The storage of the pulse is not trivial, as polarization
consists of a combination of electric-dipole and electric-quadrupole modes.
The polarization will, in general, be a mix of $x$ and $y$ polarization with a
different direction on each atom (see Ref. [22] Supplemental Material).
However, at the time of maximum mode occupation, $\approx 0.75$ of the
incident energy has been transferred to the atomic excitations.

##  VI Concluding remarks

Our protocols provide full control over the directional absorption, storage,
and emission of a single-photon pulse. Light incident from the forward
direction, backward direction, or an arbitrary combination can first be
absorbed and stored by coupling the in-plane to the out-of-plane modes. The
excitation can then be transferred to any desired combination of spatially
delocalized out-of-plane collective modes by coupling symmetric and
antisymmetric modes, e.g., to store it in the more subradiant antisymmetric
mode $\mathcal{P}_{xa}$. After some time, it can then be transferred to a
superposition of the antisymmetric and symmetric out-of-plane modes
$\mathcal{P}_{xa}$ and $\mathcal{P}_{xs}$, and, by choosing a suitable release
time $T_{r}$, can be emitted in an arbitrarily weighted superposition of
forward and backward directions, independent of that of the initial incident
photon pulse. Such efficient and controllable absorption, storage, and
emission could be utilized to take advantage of the effective 1D emission of
atomic lattices, as part of a larger architecture, to transfer excitations
forward and backward between layers for processing or storage. Additional
control of the direction, phase, and polarization of the emitted photon could
also be achieved with more complex, spatially varying level shifts [77], with
the system obtaining more general properties of highly sought after quantum
metasurfaces [78].

###### Acknowledgements.

We acknowledge financial support from the United Kingdom Engineering and
Physical Sciences Research Council (EPSRC) (Grants No. EP/S002952/1, No.
EP/P026133/1, and No. EP/W005638/1).

##  Appendix A Four-mode model

In the main text we make use of a four-mode model to explain the dynamics in
terms of symmetric and antisymmetric in-plane and out-of-plane modes, with
various couplings between them. Here, we illustrate how these couplings arise.

We first consider coupling the in-plane component to the out-of-plane
component. For a single atom at $\mathbf{r}=0$ in the circular polarization
basis the polarization amplitudes evolve as

$\partial_{t}\mathcal{P}_{\mu}=(i\Delta_{\mu}+i\delta\Omega-\gamma)\mathcal{P}_{\mu}+\zeta_{\mu}(t),$
(12)

with
$\zeta_{\mu}(t)=i(\xi/\mathcal{D})\hat{\mathbf{{e}}}_{\mu}^{\ast}\cdot\epsilon_{0}\bm{\mathbf{\cal
E}}(0,t)$. Writing these amplitudes in the Cartesian basis gives

$\displaystyle\partial_{t}\mathcal{P}_{x}$
$\displaystyle=(i\tilde{\delta}+i\delta\Omega-\gamma)\mathcal{P}_{x}+\bar{\delta}\mathcal{P}_{y},$
$\displaystyle\partial_{t}\mathcal{P}_{y}$
$\displaystyle=(i\tilde{\delta}+i\delta\Omega-\gamma)\mathcal{P}_{y}-\bar{\delta}\mathcal{P}_{x}+\zeta_{y}(t),$

with $\tilde{\delta}=(\Delta_{+}+\Delta_{-})/2$, and
$\bar{\delta}=(\Delta_{+}-\Delta_{-})/2$ coupling the $x$ and $y$ polarization
on a single atom, and where the $y$ polarized incident field propagating in
the $x$ direction [see Eq. (5)] drives only $\mathcal{P}_{y}$. The same
principle of coupling can be generalized to a single 2D array of atoms in the
$yz$ plane in which case the dipoles oscillating in phase in the $y$ direction
correspond to the in-plane mode and the dipoles oscillating in phase in the
$x$ direction correspond to the out-of-plane mode [4, 33]. Similarly for the
full bilayer lattice the in-plane and out-of-plane modes of the unperturbed
lattice are no longer eigenmodes of the system in the presence of Zeeman
splitting but are coupled together. Coupling between, e.g., the symmetric
modes can be described by a two-mode model in analogy with the single-atom
case,

$\begin{pmatrix}\dot{\mathcal{P}}_{xs}\\\
\dot{\mathcal{P}}_{ys}\end{pmatrix}=i\begin{pmatrix}\lambda_{xs}+\delta\Omega+\tilde{\delta}&-i\bar{\delta}\\\
i\bar{\delta}&\lambda_{ys}+\delta\Omega+\tilde{\delta}\end{pmatrix}\begin{pmatrix}\mathcal{P}_{xs}\\\
\mathcal{P}_{ys}\end{pmatrix}+\begin{pmatrix}0\\\ \zeta_{ys}\end{pmatrix},$

where $\lambda_{n}=\delta_{n}+i\upsilon_{n}$ and

$\zeta_{ys}=i\sqrt{2}(\xi/\mathcal{D})\epsilon_{0}\mathcal{N}h(t)\cos{(ka/2)}.$
(13)

An identical equation describes coupling between $\mathcal{P}_{ya}$ and
$\mathcal{P}_{xa}$, with $\lambda_{ys}$, $\lambda_{xs}$ replaced with
$\lambda_{ya}$, $\lambda_{xa}$ and $\zeta_{ys}$ replaced with

$\zeta_{ya}=-\sqrt{2}(\xi/\mathcal{D})\epsilon_{0}\mathcal{N}h(t)\sin{(ka/2)}.$
(14)

Coupling between, e.g., the symmetric and antisymmetric in-plane modes is most
easily explained by first considering the regime $\lambda\lesssim
a\ll\sqrt{A}$, where $A$ is the area of each layer. In this regime analytic
expressions can be found for linewidths and the relative line shifts of the
modes. However, the case $a<\lambda$ is described by identical couplings
between modes, but with $\lambda_{ya}$, $\lambda_{ys}$ replaced with values
found from full numerical analysis.

For $\lambda\lesssim a\ll\sqrt{A}$, light propagation between the layers is
effectively 1D with each lattice responding as a single “superatom” [33, 16].
We write the modes in terms of the uniform in-plane collective excitations
$\mathcal{P}_{y1}$ of layer 1 (at $x=-a/2$) and $\mathcal{P}_{y2}$ of layer 2
(at $x=a/2$), respectively, as

$\displaystyle\mathcal{P}_{ys}=\frac{1}{\sqrt{2}}\left(\mathcal{P}_{y1}+\mathcal{P}_{y2}\right),$
(15)
$\displaystyle\mathcal{P}_{ya}=\frac{1}{\sqrt{2}}\left(\mathcal{P}_{y1}-\mathcal{P}_{y2}\right).$
(16)

We apply uniform level shifts $\Delta^{(j)}_{\mu}=\Delta^{(1)}$ for $j\leq
N/2$, with $N$ total atoms ordered such that $j\leq N/2$ refers to the bottom
($x<0$) layer, and $\Delta^{(j)}_{\mu}=\Delta^{(N)}$ for $j>N/2$, referring to
the top ($x>0$) layer. Then the amplitudes of each layer evolve as

$\begin{pmatrix}\dot{\mathcal{P}}_{y1}\\\
\dot{\mathcal{P}}_{y2}\end{pmatrix}=i\begin{pmatrix}\lambda_{\rm
1D}+\delta\Omega+\Delta^{(1)}&i\mathsf{G}_{12}\\\
i\mathsf{G}_{12}&\lambda_{\rm
1D}+\delta\Omega+\Delta^{(N)}\end{pmatrix}\begin{pmatrix}\mathcal{P}_{y1}\\\
\mathcal{P}_{y2}\end{pmatrix}\\\ +\begin{pmatrix}\zeta_{y1}\\\
\zeta_{y2}\end{pmatrix},$

where $\lambda_{\rm 1D}=\delta_{\rm 1D}+i\upsilon_{\rm 1D}$ consists of a
collective level shift $\delta_{\rm 1D}$ and linewidth $\upsilon_{\rm
1D}=3\pi\gamma/(k^{2}d^{2})$ of the uniformly excited array [69], and
$\mathsf{G}_{12}=\upsilon_{\rm 1D}\exp{(ika)}$ is an effective 1D interaction
between the layers [73]. Returning to the basis of $\mathcal{P}_{ya}$,
$\mathcal{P}_{ys}$, these are no longer eigenmodes but are coupled together,

$\partial_{t}\begin{pmatrix}\mathcal{P}_{ya}\\\
\mathcal{P}_{ys}\end{pmatrix}=i\begin{pmatrix}\lambda_{ya}+\delta\Omega+\tilde{\delta}&&\Delta_{a,s}\\\
\Delta_{a,s}&&\lambda_{ys}+\delta\Omega+\tilde{\delta}\end{pmatrix}+\begin{pmatrix}\zeta_{ya}\\\
\zeta_{ys}\end{pmatrix},$

with $\lambda_{ya}=\lambda_{\rm 1D}-i\mathsf{G}_{12}$,
$\lambda_{ys}=\lambda_{\rm 1D}+i\mathsf{G}_{12}$,
$\tilde{\delta}=(\Delta^{(1)}+\Delta^{(N)})/2$, and
$\Delta_{a,s}=(\Delta^{(1)}-\Delta^{(N)})/2$.

Now we take the level shifts to be equal in each plane of the bilayer array
(no variation in the $yz$ plane), with both a possible shift between the
planes (along the $x$ direction) and a possible Zeeman splitting
$\Delta_{+}^{(j)}\neq\Delta_{-}^{(j)}$ between the $m=\pm 1$ levels of each
atom. Combining the transformation to Cartesian coordinates with that to a
diagonal basis of $\mathcal{H}$, the full control can then be described by a
four-mode model;

$\partial_{t}\begin{pmatrix}\mathcal{P}_{xa}\\\ \mathcal{P}_{ya}\\\
\mathcal{P}_{xs}\\\
\mathcal{P}_{ys}\end{pmatrix}=i(\Lambda+\delta\Omega\mathbbm{1}+\delta\mathcal{H})\begin{pmatrix}\mathcal{P}_{xa}\\\
\mathcal{P}_{ya}\\\ \mathcal{P}_{xs}\\\
\mathcal{P}_{ys}\end{pmatrix}+\begin{pmatrix}0\\\ \zeta_{ya}\\\ 0\\\
\zeta_{ys}\end{pmatrix},$ (17)

with
$\Lambda=\mathrm{diag}(\lambda_{xa},\lambda_{ya},\lambda_{xs},\lambda_{ys})$
for $\lambda_{n}=\delta_{n}+i\upsilon_{n}$ and

$\delta\mathcal{H}=\begin{pmatrix}\tilde{\delta}&-i\Delta_{x,y}&\Delta_{a,s}&-i\Delta_{a,s;x,y}\\\
i\Delta_{x,y}&\tilde{\delta}&i\Delta_{a,s;x,y}&\Delta_{a,s}\\\
\Delta_{a,s}&-i\Delta_{a,s;x,y}&\tilde{\delta}&-i\Delta_{x,y}\\\
i\Delta_{a,s;x,y}&\Delta_{a,s}&i\Delta_{x,y}&\tilde{\delta}\end{pmatrix},$
(18)

where

$\displaystyle\tilde{\delta}$
$\displaystyle=(\Delta_{+}^{(1)}+\Delta_{-}^{(1)}+\Delta_{+}^{(N)}+\Delta_{-}^{(N)})/4,$
(19) $\displaystyle\Delta_{x,y}$
$\displaystyle=(\Delta_{+}^{(1)}-\Delta_{-}^{(1)}+\Delta_{+}^{(N)}-\Delta_{-}^{(N)})/4,$
(20) $\displaystyle\Delta_{a,s}$
$\displaystyle=(\Delta_{+}^{(1)}+\Delta_{-}^{(1)}-\Delta_{+}^{(N)}-\Delta_{-}^{(N)})/4,$
(21) $\displaystyle\Delta_{a,s;x,y}$
$\displaystyle=(\Delta_{+}^{(1)}-\Delta_{-}^{(1)}-\Delta_{+}^{(N)}+\Delta_{-}^{(N)})/4.$
(22)

Hence, as well as an overall average level shift $\tilde{\delta}$,
$\Delta_{x,y}$ couples in-plane modes to out-of-plane modes while preserving
symmetry, $\Delta_{a,s}$ couples symmetric modes to antisymmetric modes while
preserving direction, and $\Delta_{a,s;x,y}$ couples the symmetric in-plane to
antisymmetric out-of-plane and vice-versa. For example, when a photon pulse is
stored in the out-of-plane modes and the shifts between the $m=\pm 1$ levels
are turned off, $\Delta_{a,s}$ is the only non-zero off-diagonal element in
Eq. (18), coupling $\mathcal{P}_{xs}$ and $\mathcal{P}_{xa}$. This can be
utilized in the control of the superposition of the stored photon and, hence,
the emission direction of the photon upon retrieval, as described in Sec.
IV.4.

Going beyond the regime $\lambda\lesssim a\ll\sqrt{A}$ and considering
$a<\lambda$ is straightforward: The effective parameters described in Eqs.
(19-22) lead to identical coupling between modes, as described in Eq. (18),
with $\Lambda$ found by numerically diagonalizing the full $3N\times 3N$
matrix $\mathcal{H}$ in Eq. (1). We then choose the eigenmodes which most
closely resemble the ideal uniform symmetric and antisymmetric in-plane and
out-of-plane modes.

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