# Chiral Quantum-Optical Elements for Waveguide-QED with Sub-wavelength
Rydberg-Atom Arrays

Lida Zhang (张理达)1 Fan Yang (杨帆)2 Klaus Mølmer2 Thomas Pohl3 1School of
Physics, East China University of Science and Technology, Shanghai, 200237,
China  2Center for Hybrid Quantum Networks, Niels Bohr Institute, University
of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark  3Institute for
Theoretical Physics, TU Wien, Wiedner Hauptstraße 8-10/136, A-1040 Vienna,
Austria

###### Abstract

We describe an approach to achieve near-perfect unidirectional light-matter
coupling to an effective quantum emitter that is formed by a subwavelength
array of atoms in the Rydberg-blockade regime. The nonlinear reflection and
transmission of such two-dimensional superatoms are exploited in different
interferometric setups for the deterministic generation of tunable single
photons and entangling two-photon operations with high fidelities,
$\mathcal{F}\gtrsim 0.999$. The described setup can function as a versatile
nonlinear optical element in a free-space photonic quantum network with simple
linear elements and without the need of additional mode confinement, optical
resonators, or optical isolators.

##  I Introduction

While the use of photons for long-distance communication is well established,
their application in quantum technologies for quantum computing Slussarenko
and Pryde (2019); Bartlett  _et al._ (2021), simulation Aspuru-Guzik and
Walther (2012); Hartmann (2016), and metrology Giovannetti  _et al._ (2011);
Polino  _et al._ (2020) has been rapidly advancing in recent years. All such
applications typically require a light-matter interface with a nonlinear
optical response in order to generate and manipulate nonclassical states of
light. Realizing the strongest possible nonlinearity that acts at the level of
single light-quanta would enable the deterministic processing of photonic
quantum information. These prospects have motivated substantial research
efforts towards reaching this extreme limit of quantum nonlinear optics Chang
_et al._ (2014), using single atoms trapped in high-finesse cavities Birnbaum
_et al._ (2005); Reiserer and Rempe (2015); Welte  _et al._ (2018), quantum
dots coupled to photonic nanostructures Tiecke  _et al._ (2014); Petersen  _et
al._ (2014); Lodahl  _et al._ (2015); Chang  _et al._ (2018); Yu  _et al._
(2019); Prasad  _et al._ (2020), or ensembles of strongly interacting Rydberg
atoms Peyronel  _et al._ (2012); Paris-Mandoki  _et al._ (2017a); Murray and
Pohl (2016); Firstenberg  _et al._ (2016); Thompson  _et al._ (2017); Yang
_et al._ (2020). While strong optical nonlinearities have been demonstrated in
these systems, the suppression of decoherence and photon loss to enable the
high-fidelity control and manipulation of single-photon quantum states remains
an outstanding challenge.

Coherent light-matter interfacing, on the other hand, can be achieved with
two-dimensional atomic arrays Facchinetti  _et al._ (2016); Bettles  _et al._
(2016); Shahmoon  _et al._ (2017), in which photon scattering is suppressed
collectively for subwavelength spacing between the atoms. The cooperative
response of two-dimensional emitter arrays enables applications as linear
optical elements, e.g, for efficient optical memories and for high-fidelity
wavefront shaping Facchinetti  _et al._ (2016); Perczel  _et al._ (2017);
Guimond  _et al._ (2019); Ballantine and Ruostekoski (2020); Grankin  _et al._
(2018); Baßler  _et al._ (2023). The strong collective photon coupling also
permits to generate strong nonlinearities by using multiple arrays Pedersen
_et al._ (2023) or by exploiting the strong interaction between atomic Rydberg
states in the array Bekenstein  _et al._ (2020); Moreno-Cardoner  _et al._
(2021); Zhang  _et al._ (2022).

Figure 1: (a) Illustration of a photonic network in which arrays of Rydberg
atoms form nonlinear optical elements (NLOE) that are capable of generating
tuneable single photons and performing entangling two-photon operations.
Illuminating the array at a finite angle offers a simple means for
unidirectional coupling whereby an incident quantum state in an input mode
($\hat{a}_{\rightarrow}^{({\rm in})}$) is transferred into to two output
modes, $\hat{a}_{\rightarrow}$ and $\hat{a}_{\leftarrow}$, that all propagate
along the same direction. Such a mode-selective coupling is possible with a
high efficiency $\beta\sim 1$, which is shown in (c) as function of the
principal quantum number of $nS$-states of Rubidium atoms in an array with
subwavelength spacing of $a=0.65\lambda$.

Here, we combine these features and propose a complete toolbox for the high-
fidelity deterministic generation and processing of photonic quantum states,
based on sub-wavelength arrays of three-level Rydberg atoms. We identify
optimal driving schemes and parameters that yield strong single-mode photon-
coupling with virtually no losses, while maximizing the effect of Rydberg-
Rydberg atom interactions. The extended geometry of the array can be used to
achieve unidirectional photon coupling, and thereby approach the ideal limit
of chiral waveguide-QED Sheremet  _et al._ (2023), in which a single saturable
quantum emitter is perfectly coupled to a single propagating photonic mode
without any scattering losses. Detailed calculations for finite arrays that
include relevant loss and decoherence sources show that unidirectional
coupling efficiencies of $\beta\sim 0.9993$ should be achievable with typical
Rydberg states of ongoing experiments [see Fig. 1(c)]. We show that this
system can work as a versatile nonlinear optical element that offers all
functionalities needed in a photonic quantum network, from the generation of
tunable single-photon pulses to high-fidelity two-photon operations [see Figs.
1(a) and 1(b)], such as photon sorting and conditional phase-gates.
Importantly, the Rydberg atom array is intrinsically mode-matched to any
incident transverse photonic mode and it can realize nearly perfect entangling
two-photon operations with infidelities below the per mille level.

The article is organized as follows. In Sec. II, we describe the basic setup
of three-level Rydberg-atom arrays and discuss their linear optical response
to an incident continuous wave (cw)-field. In Sec. III, we outline a
simplified description of the array in terms of two-level emitters and discuss
conditions to obtain narrow reflection resonances and strong interactions. The
optical nonlinearity that results from the Rydberg-blockade between excited
atoms is characterized through emerging two-photon correlations in Sec. IV. In
section V, we propose a setup to implement unidirectional photon coupling and
we consider the interaction with propagating photon pulses in the ensuing
section VI. The application of this setup to generate single photons and
perform two-photon operations is detailed in Sec. VII along with a
characterization of its performance for these tasks.

##  II Three-level Rydberg-atom arrays

The nonlinear optical elements discussed above are formed by two-dimensional
arrays of $N$ three-level atoms at positions ${\bf r}_{i}$, $i=1,...,N$. The
level scheme of the atoms is illustrated in Fig.2(a). The incident probe light
is described by the quantum field $\hat{\mathcal{E}}(\mathbf{r})$ that
satisfies bosonic commutation relations,
$[\hat{\mathcal{E}}(\mathbf{r}),\hat{\mathcal{E}}^{\dagger}(\mathbf{r}^{\prime})]=\delta(\mathbf{r}-\mathbf{r}^{\prime})$,
and defines the spatial photon density as
$\langle\hat{\mathcal{E}}^{\dagger}(\mathbf{r})\hat{\mathcal{E}}(\mathbf{r})\rangle$.
The quantum field drives the transition between the ground state $|g\rangle$
and excited state $|e\rangle$ of each atom in the array. The probe light
excites the atoms with a frequency detuning $\Delta_{e}$ and coupling strength
$g=\lambda\sqrt{3\Gamma c/(8\pi)}$ that is determined by the decay rate
$\Gamma$ and resonant wavelength $\lambda$ of the $|e\rangle$-$|g\rangle$
transition. In addition, a classical control laser field couples the state
$|e\rangle$ to a high-lying Rydberg state $|r\rangle$ with a corresponding
control Rabi frequency $\Omega$ and frequency detuning $\Delta_{r}$. The
Hamiltonian

$\displaystyle\hat{H}_{0}=$
$\displaystyle-\sum^{N}_{j=1}\big{(}\Delta_{e}\hat{\sigma}^{(j)}_{ee}+(\Delta_{e}+\Delta_{r})\hat{\sigma}^{(j)}_{rr}$
$\displaystyle\quad\quad\quad+[g\hat{\mathcal{E}}(\mathbf{r}_{j})\hat{\sigma}^{(j)}_{eg}+\Omega\hat{\sigma}^{(j)}_{re}+h.c.]\big{)}$
(1)

describes the coupling of the atoms to the two applied light fields within the
rotating wave approximation.

Moreover, the atoms interact with the free-space radiation field, whereby
multiple scattering of photons on the $|g\rangle-|e\rangle$ transition of
atoms across the array gives rise to collective light-matter interactions.
Such processes can be described effectively by integrating out the photonic
dynamics within the Born-Markov approximation Gross and Haroche (1982); Dung
_et al._ (2002); Ficek and Tanaś (2002). This yields a master equation

$\displaystyle\partial_{t}\hat{\rho}=-i\left[\hat{H}_{0}+\hat{H}_{\rm
dd},\hat{\rho}\right]+\mathcal{L}_{\rm dd}(\hat{\rho})+\mathcal{L}_{\rm
Ryd}(\hat{\rho})$ (2)

for the many-body density matrix of the atoms, where

$\displaystyle\hat{H}_{\text{dd}}=$ $\displaystyle-\sum^{N}_{j,k\neq
j}J_{jk}\hat{\sigma}^{(j)}_{eg}\hat{\sigma}^{(k)}_{ge}$ (3)

describes the coherent exchange of atomic excitations due to emission and re-
absorption of photons, while the Lindblad operator

$\displaystyle\mathcal{L}_{\rm dd}(\rho)=$
$\displaystyle\sum^{N}_{j,k}\frac{\Gamma_{jk}}{2}(2\hat{\sigma}^{(k)}_{ge}\rho\hat{\sigma}^{(j)}_{eg}-\\{\hat{\sigma}^{(j)}_{eg}\hat{\sigma}^{(k)}_{ge},\rho\\})$
(4)

captures the dynamics of corresponding collective decay processes that emerges
from the light-induced dipole-dipole interactions between the atoms. The
interaction coefficients

$\displaystyle
J_{jk}+i\frac{\Gamma_{jk}}{2}=\frac{\mu_{0}\omega^{2}_{eg}}{\hbar}{\bm{d}}^{*}_{eg}{\bm{G}}(\mathbf{r}_{j},\mathbf{r}_{k},\omega_{eg}){\bm{d}}_{eg}$
(5)

are related to the Green’s function tensor, $\bm{G}$, of the free-space
electromagnetic field at the resonance frequency, $\omega_{eg}=2\pi
c/\lambda$, and to the transition dipole moment $\bm{d}_{eg}$ of the
$|e\rangle-|g\rangle$ transition. While the transition dipole moment of the
$|e\rangle-|r\rangle$ Rydberg-state transition is too small to cause
significant atomic excitation-exchange interactions, we include the
spontaneous decay of the Rydberg-state with a rate $\gamma$ via the
Lindbladian

$\displaystyle\mathcal{L}_{\rm Ryd}(\rho)=$
$\displaystyle\sum_{j}\frac{\gamma}{2}(2\hat{\sigma}^{(j)}_{er}\hat{\rho}\hat{\sigma}^{(j)}_{re}-\\{\hat{\sigma}^{(j)}_{re}\hat{\sigma}^{(j)}_{er},\hat{\rho}\\}).$
(6)

Having obtained the solution of Eq.(2) for the density matrix of the atoms, we
can reconstruct the quantum state of the light scattered by the atomic array.
Let’s consider a single-mode input field, whereby all incident photons occupy
a single spatial mode $u_{\rightarrow}({\bf r})$ or $u_{\leftarrow}({\bf r})$
that obeys the paraxial wave equation $[4\pi
i\partial_{z}+\lambda(\partial_{x}^{2}+\partial_{y}^{2})+8\pi^{2}\lambda^{-1}]u_{\rightarrow}=0$
and $[4\pi
i\partial_{z}-\lambda(\partial_{x}^{2}+\partial_{y}^{2})-8\pi^{2}\lambda^{-1}]u_{\leftarrow}=0$
for forward ($u_{\rightarrow}$) and backward ($u_{\leftarrow}$) propagating
light. This equation describes the propagation of light along the $z$-axis and
the operators $\hat{a}^{({\rm in})}_{\rightarrow}=\int{\rm d}{\bf
r}\>u^{*}_{\rightarrow}({\bf r})\hat{\mathcal{E}}({\bf r})$ and
$\hat{a}^{({\rm in})}_{\leftarrow}=\int{\rm d}{\bf r}\>u^{*}_{\leftarrow}({\bf
r})\hat{\mathcal{E}}({\bf r})$ shall define the occupation of photons in the
incident field, propagating in the forward and backward direction,
respectively. One can then obtain simple input-output relations Manzoni  _et
al._ (2018)

$\displaystyle\hat{a}_{\rightarrow}(t)=$
$\displaystyle\hat{a}^{(\text{in})}_{\rightarrow}(t)+i\frac{g}{c}\sqrt{\ell}\sum_{j}u^{*}(\mathbf{r}_{j})\hat{\sigma}_{ge}^{(j)}(t),$
(7a) $\displaystyle\hat{a}_{\leftarrow}(t)=$
$\displaystyle\hat{a}^{(\text{in})}_{\leftarrow}(t)+i\frac{g}{c}\sqrt{\ell}\sum_{j}u^{*}(\mathbf{r}_{j})\hat{\sigma}_{ge}^{(j)}(t)$
(7b)

for the total light field in the modes $u_{\rightarrow}$ and $u_{\leftarrow}$,
with $u_{\rightarrow}({\bf r}_{j})=u_{\leftarrow}({\bf r}_{j})=u({\bf
r}_{j})/\sqrt{\ell}$ in the plane of the array. Here, $\ell^{-1}=\int{\rm
d}{\bf r}_{\perp}|u_{\rightarrow}({\bf r})|^{2}$ defines the inverse length of
the quantization volume for the free-space field modes along the propagation
direction, such that the transverse mode function is normalized as $\int{\rm
d}{\bf r}_{\perp}|u({\bf r})|^{2}=1$, where we have used ${\bf
r}_{\perp}=(x,y)$ for the transverse coordinates. We consider coherent driving
such that one can replace $\hat{\mathcal{E}}$ by
$\mathcal{E}=\langle\hat{a}_{\rightarrow}^{({\rm in)}}\rangle
u_{\rightarrow}({\bf r})+\langle\hat{a}_{\leftarrow}^{({\rm in)}}\rangle
u_{\leftarrow}({\bf r})$, which defines the probe Rabi frequency
$\Omega_{p}({\bf r})=g\mathcal{E}({\bf r})$. The operators
$\hat{a}_{\rightarrow}$ and $\hat{a}_{\leftarrow}$, respectively, describe the
transmitted ($z>0$) and reflected ($z<0$) photons in the incident spatial
modes. For an exclusively forward propagating input field
($\langle\hat{a}^{\dagger({\rm in})}_{\leftarrow}\hat{a}^{({\rm
in})}_{\leftarrow}\rangle=0$), we can define the reflectivity
$R=\langle\hat{a}^{\dagger}_{\leftarrow}\hat{a}_{\leftarrow}\rangle/\langle\hat{a}^{\dagger({\rm
in})}_{\rightarrow}\hat{a}^{({\rm in})}_{\rightarrow}\rangle$ and transmission
$T=\langle\hat{a}^{\dagger}_{\rightarrow}\hat{a}_{\rightarrow}\rangle/\langle\hat{a}^{\dagger({\rm
in})}_{\rightarrow}\hat{a}^{({\rm in})}_{\rightarrow}\rangle$ of photons in
the mode $u_{\rightarrow}({\bf r})$ and $u_{\leftarrow}({\bf r})$.
Correspondingly, the loss coefficient is given by $L=1-T-R$ and accounts for
the scattering of photons into other transversal modes.

Figure 2: (a) Illustration of the atomic three-level system and its
approximation by a two-level configuration. (b) Comparison of the steady-state
linear reflection spectrum as a function of detuning $\Delta_{e}$ determined
by the full three-level model and the effective two-level approximation. (c)
Detailed comparison of the linear transmission (T), reflection (R), and loss
(L) spectrum around the right absorption peak. The dots and the solid lines
are obtained with the three-level configuration and the effective two-level
approximation, respectively. Here a rounded $21\times 21$ array is chosen,
where atoms in the corners are removed to form a circular disc with an atom
number of 317. The remaining parameters are
$\Omega_{p}(\mathbf{0})/\Gamma=0.001,w_{0}=3.0\lambda$, and
$\Omega/\Gamma=8.0,a=0.75\lambda,\Delta_{r}/\Gamma=-10.0$.

For a single incident photon, Eq.(2) only involves a single atomic excitation
and can be readily solved. The resulting expression Bekenstein  _et al._
(2020); Zhang  _et al._ (2022)

$\displaystyle
R=\left|\frac{g^{2}}{ca^{2}}\frac{1}{\Delta_{e}+\Delta_{c}-\frac{\Omega^{2}}{\Delta_{e}+\Delta_{r}+i\gamma/2}+i\frac{\Gamma_{c}}{2}}\right|^{2},$
(8)

for the reflection coefficient involves the collective level shift,
$\Delta_{c}$, and collective decay rate, $\Gamma_{c}$, of the atomic array
respectively, which arise from the dipole-dipole interaction in Eqs.(3) and
(4). For large atomic lattices and broad input beams, the collective Lamb
shift is given by $\Delta_{c}=\sum_{j\neq i}J_{i,j}$, while the collective
decay rate approaches $\Gamma_{\rm
c}=\frac{3}{4\pi}\frac{\lambda^{2}}{a^{2}}\Gamma=\frac{2g^{2}}{ca^{2}}$ for
high atom numbers, $N\rightarrow\infty$, and large beam waists,
$w_{0}\gg\lambda$.

Equation (8) is valid for an infinitely extended array and a vanishing
transverse photon momentum across the array. In the numerical calculations we
consider incident beams with a Gaussian mode profile

$u({\bf r}_{\perp},z=0)=\sqrt{\frac{2}{\pi
w_{0}^{2}}}\exp\left(-\frac{r_{\perp}^{2}}{w_{0}^{2}}\right)$ (9)

where ${\bf r}_{\perp}=(x,y)$, $z=0$ defines the position within the plane of
the array, and $w_{0}$ denotes the Gaussian waist of the beam. We consider
square lattices, where all atoms outside a given radius are removed to form a
circularly shaped array. This reduces the atom number at a given system size
and thereby facilitates the numerical simulations of larger lattices.

A typical reflection spectrum obtained from such a calculation is shown in
Fig.2(b). The spectrum features two Autler-Townes reflection peaks and a
minimum around two-photon resonance, $\Delta_{e}+\Delta_{r}=0$, where
electromagnetically induced transparency Fleischhauer  _et al._ (2005)
enhances transmission. At this point transmission is only limited by Rydberg-
state decay which yields a small reflection coefficient
$R\approx(\frac{g^{2}}{ca^{2}}\frac{\gamma}{2\Omega^{2}})^{2}$ that vanishes
with increasing principal quantum number $n$ as the Rydberg-state decay rate,
$\gamma$, decreases. In addition one finds two reflection resonances at the
probe detunings

$\Delta_{e}^{(\pm)}=[-(\Delta_{r}+\Delta_{c})\pm\sqrt{(\Delta_{r}-\Delta_{c})^{2}+4\Omega^{2}})]/2,$
(10)

which are determined by the Rydberg-state detuning $\Delta_{r}$ and the
collective level shift $\Delta_{c}$. The two probe detunings
$\Delta_{e}=\Delta_{e}^{(\pm)}$ correspond to the Autler-Townes resonances of
the three-level atoms, which become well separated for large control Rabi
frequencies $\Omega\gg\Gamma_{c}$.

##  III Reflection resonances and two-level limit

The two Autler-Townes resonances correspond to the energies of the dressed
two-level system, formed by the two atomic states $|e\rangle$ and $|r\rangle$
that are coupled by the control-field Rabi frequency $\Omega$ [see Fig.2(a)].
For large Rydberg-state detunings $\Delta_{r}$, this coupling is far off
resonant such that state-mixing is small. Consequently, one of the two dressed
states is primarily composed of the intermediate state $|e\rangle$, which
yields a broad reflection resonance, while the other predominantly consists of
the Rydberg-state state $|r\rangle$ and generates a narrow reflection
resonance. Hence, the narrow resonance is most sensitive to Rydberg-Rydberg
atom interactions and therefore yields ideal conditions to exploit the Rydberg
blockade for generating large optical nonlinearities.

For large Rabi frequencies $\Omega$ and correspondingly large values of
$\Delta_{e}$, one can adiabatically eliminate the $|e\rangle$-state dynamics
around the narrow Autler-Townes resonance. This yields an approximate master
equation

$\displaystyle\partial_{t}\hat{\rho}=-i\left[\hat{H},\hat{\rho}\right]+\mathcal{L}(\hat{\rho})$
(11)

for the driven dynamics of the ground and Rydberg state with an effective two-
level Hamiltonian (see Appendix A)

$\displaystyle\hat{H}\approx$
$\displaystyle-\sum_{i}\bar{\Delta}\hat{\sigma}^{(i)}_{rr}+[\bar{g}\hat{\mathcal{E}}(\mathbf{r}_{i})\hat{\sigma}^{(i)}_{rg}+h.c.]-\sum_{j\neq
i}\bar{J}_{ij}\hat{\sigma}^{(i)}_{gr}\hat{\sigma}^{(j)}_{rg},$

and Lindblad operator

$\displaystyle\mathcal{L}(\rho)=$
$\displaystyle\sum_{i,j}\frac{\bar{\Gamma}_{ij}+\gamma\delta_{ij}}{2}(2\hat{\sigma}^{(j)}_{gr}\rho\hat{\sigma}^{(i)}_{rg}-\\{\hat{\sigma}^{(i)}_{rg}\hat{\sigma}^{(j)}_{gr},\rho\\}).$
(13)

As indicated in Fig.2(b), the effective frequency detuning, single-photon
coupling strength and induced dipole-dipole interactions are, respectively,
given by $\bar{\Delta}=\Delta_{e}+\Delta_{r}-\frac{\Omega^{2}}{\Delta_{e}}$,
$\bar{g}=-g\Omega/\Delta_{e}$, $\bar{J}_{ij}=J_{ij}\Omega^{2}/\Delta^{2}_{e}$,
and $\bar{\Gamma}_{ij}=\Gamma_{ij}\Omega^{2}/\Delta^{2}_{e}$. Similarly, the
photon field can now be obtained as (see Appendix A)

$\displaystyle\hat{a}_{\rightarrow}(t)=$
$\displaystyle\hat{a}^{(\text{in})}_{\rightarrow}(t)+i\frac{\bar{g}}{c}\sqrt{\ell}\sum_{j}u^{*}(\mathbf{r}_{j})\hat{\sigma}_{gr}^{(j)}(t),$
(14a) $\displaystyle\hat{a}_{\leftarrow}(t)=$
$\displaystyle\hat{a}^{(\text{in})}_{\leftarrow}(t)+i\frac{\bar{g}}{c}\sqrt{\ell}\sum_{j}u^{*}(\mathbf{r}_{j})\hat{\sigma}_{gr}^{(j)}(t),$
(14b)

from the collective polarization between the ground state and Rydberg state.
Figure 2(c) compares the linear transmission and reflection obtained from the
exact three-level calculation with Eqs.(III)-(14) and demonstrates excellent
agreement with the two-level approximation around the narrow Autler-Townes
resonance. Here the reflection spectrum can be written

$\displaystyle
R_{0}=\left|\frac{\bar{\Gamma}_{c}/2}{\bar{\Delta}+\bar{\Delta}_{c}+i\frac{\bar{\Gamma}_{c}+\gamma}{2}}\right|^{2},$
(15)

as a function of the detuning $\bar{\Delta}$, the small linewidth
$\bar{\Gamma}_{c}=(\Omega/\Delta_{e})^{2}\Gamma_{c}$ and the collective level
shift $\bar{\Delta}_{c}=\sum_{j\neq i}\bar{J}_{ij}$. When the photon detuning
$\Delta_{e}$ is varied over the narrow range, $\bar{\Gamma}_{c}$, of the
reflection resonance, both $\bar{\Gamma}_{c}$ and $\bar{\Delta}_{c}$ are
almost unchanged and Eq.(15) is well approximated by a Lorentzian. For a fixed
value of the ratio $\Omega/\Delta_{e}$ that controls the values of the
effective parameters in Eqs.(III) and (13), the accuracy of the effective two-
level description improves upon increasing $\Omega$ and $\Delta_{r}$, and is,
therefore, only limited by the experimentally available Rydberg-laser power.
For the parameters considered here, the reflection spectra agree to within
less than $1\%$. These conditions, yield effective two-level Rydberg-atom
mirrors with near-perfect reflection resonances for typical parameters of
current experiments with cold Rubidium Rydberg atoms Rui  _et al._ (2020);
Browaeys and Lahaye (2020); Ebadi  _et al._ (2021); Scholl  _et al._ (2021);
Semeghini  _et al._ (2021); Satzinger and et. al. (2021).

##  IV Rydberg blockade and single-photon saturation

Using this effective description in terms of Rydberg-state two-level systems,
we can now explore the effects of interactions between the excited Rydberg
states on the nonlinear optical response of the atomic array. The static
polarizability of the atoms increases rapidly with their principal quantum
number, $n$, such that Rydberg states feature strong van der Waals
interactions
$U(\mathbf{r}_{i},\mathbf{r}_{j})=C_{6}/|\mathbf{r}_{j}-\mathbf{r}_{i}|^{6}$,
with an interaction coefficient, $C_{6}\sim n^{11}$, which can exceed that of
ground state atoms by many orders of magnitude Saffman  _et al._ (2010). The
resulting level shifts of interacting pairs of Rydberg atoms can be
sufficiently large to inhibit the excitation of more than a single atom within
a diameter of several $\mu$m Lukin  _et al._ (2001); Jaksch  _et al._ (2000).
This Rydberg blockade has been demonstrated and exploited in several
experiments and a range of different applications from quantum information
processing Saffman  _et al._ (2010) to nonlinear optics Firstenberg  _et al._
(2016); Murray and Pohl (2016) and quantum simulations Browaeys and Lahaye
(2020); Wu  _et al._ (2021).

In the current setting, the Rydberg blockade causes a nonlinear reflection by
the atomic array that is highly sensitive to the number of incident photons
Dudin and Kuzmich (2012); Peyronel  _et al._ (2012); Paris-Mandoki  _et al._
(2017b); Ripka  _et al._ (2018). Under the conditions outline above, a single
resonant photon ($\bar{\Delta}=-\bar{\Delta}_{c}$) will undergo reflection
with near-unit probability, $R\sim 1$ [cf. Eq.(15)]. During this process the
photon generates a Rydberg excitation which, for a time
$\sim\bar{\Gamma}_{c}^{-1}$, blocks the coupling of other atoms to the Rydberg
state. The blockade, therefore, exposes all subsequently incident photons to
the optical response of the bare $|g\rangle-|e\rangle$ transition. The
corresponding reflection coefficient for the blockaded arrays is thus given by

$\displaystyle
R_{bl}=\left|\frac{g^{2}}{ca^{2}}\frac{1}{\Delta_{e}+\Delta_{c}+i\frac{\Gamma_{c}}{2}}\right|^{2}.$
(16)

The condition $\bar{\Delta}=-\bar{\Delta}_{c}$, for which $R_{0}\sim 1$,
implies a large single-photon detuning $|\Delta_{e}|>\Omega$ for large control
Rabi frequencies $\Omega$. Consequently, the blockade generates a low
reflection coefficient $R_{bl}<(\Gamma_{c}/\Omega)^{2}/4\ll 1$. One thus
obtains an efficient and nearly lossless mechanism for switching between near-
unit reflection and near-unit transmission by a single photon.

The blockade is effective if the van der Waals interaction exceeds the
linewidth $\bar{\Gamma}_{c}+\gamma$ of the effective two-level mirror
described by the reflection spectrum $R_{0}$, Eq.(15). This defines the
blockade radius, $r_{b}$ as

$\displaystyle\frac{C_{6}}{r_{b}^{6}}=\bar{\Gamma}_{c},$ (17)

which we show in Fig. 3(a) as a function of the principal quantum number for
$nS$-states of Rubidium atoms Li  _et al._ (2003); Beterov  _et al._ (2009).
For typical values of $n$ and $\Omega$, the narrow linewidth of the reflection
resonances yields sizeable blockade radii that significantly exceed the
wavelength $\lambda=780$nm of the atomic $D_{2}$-line in Rubidium and,
therefore, covers a large number of atoms in the subwavelength array. Such
large blockade radii permit to maintain the Rydberg blockade for probe fields
with a sufficiently large beam waist $w_{0}$ to suppress losses and
imperfections due to finite transverse momenta of the incident field.

Figure 3: (a) The Rydberg blockade radius for different principal quantum
numbers $n$ of the Rydberg-state. The steady-state correlation function
$g_{\leftleftarrows}^{(2)}(0)$ for simultaneously reflected photons is shown
in panel (b) as a function of $n$ and for different values $w_{0}$ of the
waist of the incident probe beam. The rapid drop of
$g_{\leftleftarrows}^{(2)}$ indicates the onset of an efficient blockade when
the blockade radius $r_{b}$ exceeds $w_{0}$. Parameters are $\Omega_{p}({\bf
0})/\Gamma=0.001$, $\Omega/\Gamma=8.0$, $a=0.75\lambda$, and
$\Delta_{r}/\Gamma=-10.0$.

We can employ the equal-time correlation function

$\displaystyle g_{\alpha\atop\beta}^{(2)}=$
$\displaystyle\lim_{t\rightarrow\infty}\frac{\langle\hat{a}_{\alpha}^{\dagger}(t)\hat{a}_{\beta}^{\dagger}(t)\hat{a}_{\beta}(t)\hat{a}_{\alpha}(t)\rangle}{\langle\hat{a}_{\alpha}^{\dagger}(t)\hat{a}_{\alpha}(t)\rangle\langle\hat{a}_{\beta}^{\dagger}(t)\hat{a}_{\beta}(t)\rangle}$
(18)

of the outgoing light as an optical probe to analyze the efficiency of the
Rydberg-excitation blockade. Here, the field operators describe the outgoing
photons that propagate in the forward ($\alpha,\beta=\rightarrow$) and
backward ($\alpha,\beta=\leftarrow$) direction. Figure 3(b) shows the equal-
time correlation function of simultaneously reflected photons as a function of
$n$ for different values of $w_{0}$. The value of $g_{\leftleftarrows}^{(2)}$
drops rapidly with increasing $n$ and vanishes once the blockade radius,
$r_{b}$, exceeds the waist of the Gaussian beam profile. Indeed, this
indicates a full excitation blockade of the array, leading to perfect
antibunching of the reflected light. Under these conditions, the atomic array
turns the Rydberg blockade into a photon blockade, where the inhibition of
simultaneous Rydberg excitation suppresses the simultaneous reflection of two
or more photons.

On the other hand, the transmitted light shows strong photon bunching, i.e.
$g_{\rightrightarrows}^{(2)}\gg 1$ and $g_{\leftrightarrows}^{(2)}\gg 1$ (see
Fig. 4). Since the linear transmission is highly suppressed on resonance,
significant photon transmission is only possible due to the Rydberg-blockade
induced by multiple coincident photons. Figure 4 compares the two-photon
correlations obtained from the exact three-level simulations of Eqs.(II)-(43)
with the effective two-level description according to Eqs.(III)-(14) for an
array of $7\times 7$ atoms. The results are virtually identical, and we shall
use the more efficient two-level calculations for analyses of larger systems
and wider probe beams. Under such conditions, the atomic array can act like a
giant saturable quantum emitter with near-perfect single-mode coupling to
light and very low photon losses.

Figure 4: The pair correlation function for transmitted and reflected photons
is shown as a function of the effective probe detuning $\bar{\Delta}$. The
simulation results for three-level atoms (points) agree excellently well with
approximate two-level limit (solid lines). The used parameters are
$\Delta_{r}/\Gamma=-10,\Omega/\Gamma=8$ for (a) and
$\Delta_{r}/\Gamma=-40,\Omega/\Gamma=32$ for (b), as well as
$w_{0}=1.125\lambda$, $n=100$, $\Omega_{p}({\bf 0})/\Gamma=0.001$, and
$a=0.75\lambda$. Figure 5: Beam splitter configuration to achieve
unidirectional coupling between an incident photon mode $\hat{a}_{\rm in}$ and
the atomic array. For perfect alignment one transfers the photonic input state
to a single output mode $\hat{a}$, while any signal in the second output port
$\hat{b}$ provides an interferometric measure of imperfections that may permit
fine-tuning of the setup.

##  V Unidirectional photon interactions

The collective optical response of small atomic arrays together with the
Rydberg blockade makes it possible to reach the ideal limit of waveguide-QED
Roy  _et al._ (2017); Chang  _et al._ (2018), akin to ongoing experiments with
cold atoms near nano-fibers or photonic crystals Thompson  _et al._ (2013);
Tiecke  _et al._ (2014); Petersen  _et al._ (2014), quantum dots Arcari  _et
al._ (2014); Tiranov  _et al._ (2023), or super conducting circuits Kannan
_et al._ (2020). As we have seen above, the Rydberg-atom array implements a
fully saturable quantum emitter which can be strongly coupled to a single
propagating mode with an effective coupling efficiency, $\beta$, of that can
approach record values of $\beta\sim 1$ [see Fig.1(c)], only limited by the
weak Rydberg-state decay. As opposed to point-like scatterers and waveguide-
QED, the atomic array provides a flexible optical interface that can yield
such high $\beta$-factors for any incident spatial mode.

These features can be exploited by shining the probe light at a non-orthogonal
angle, which prevents the photons from being reflected back into the initial
spatial mode [see Fig.1(a)]. The resulting splitting of the incident photons
into two distinct spatial modes can then be used to generate a directional
coupling between the photons and the array, whereby the incident light is
scattered only in the forward direction of the original input mode [see
Fig.5].

The proposed setup is illustrated in Fig. 5 and only involves a few additional
linear optical elements. First, the incident photon field, $\hat{a}^{(\rm
in)}(t)$, enters a symmetric beam splitter at one of its two input ports and
is converted into two modes, $\hat{a}^{(\rm in)}_{\rightarrow}$ and
$\hat{a}^{(\rm in)}_{\leftarrow}$, that subsequently hit the array from the
two opposite sides as indicated in Fig. 5. The interaction with the array
generates the following output fields

$\displaystyle\hat{a}_{\rightarrow}(t)=$ $\displaystyle\hat{a}^{(\rm
in)}_{\rightarrow}(t)+\frac{ig}{c}\sqrt{\ell}\sum_{j}u^{*}(\mathbf{r}_{j})\hat{\sigma}^{(j)}_{ge},$
(19a) $\displaystyle\hat{a}_{\leftarrow}(t)=$ $\displaystyle\hat{a}^{(\rm
in)}_{\leftarrow}(t)+\frac{ig}{c}\sqrt{\ell}\sum_{j}u^{*}(\mathbf{r}_{j})\hat{\sigma}^{(j)}_{ge},$
(19b)

where the mode function $u(\mathbf{r}_{j})$ depends on the angle of incidence
$\theta$, but is identical for $\hat{a}_{\rightarrow}^{(\rm in)}$ and
$\hat{a}_{\leftarrow}^{(\rm in)}$ under perfectly symmetric conditions.
Finally, a second beam splitter is used to recombine the two fields, and
yields

$\displaystyle\hat{a}(t)=$ $\displaystyle\frac{\hat{a}^{(\rm
in)}_{\rightarrow}(t)+\hat{a}^{(\rm
in)}_{\leftarrow}(t)}{\sqrt{2}}+\frac{i\sqrt{2}g}{c}\sqrt{\ell}\sum_{j}u^{*}(\mathbf{r}_{j})\hat{\sigma}^{(j)}_{ge}$
(20)

on one of its output ports, and

$\displaystyle\hat{b}(t)=$ $\displaystyle\frac{\hat{a}^{(\rm
in)}_{\rightarrow}(t)-\hat{a}^{(\rm in)}_{\leftarrow}(t)}{\sqrt{2}}$ (21)

on the other. Overall, this converts a single forward-propagating input field
$\hat{a}^{(\rm in)}(t)$ into a single output field

$\displaystyle\hat{a}(t)=$ $\displaystyle\hat{a}^{(\rm
in)}(t)+\frac{i\sqrt{2}g}{c}\sqrt{\ell}\sum_{j}u^{*}(\mathbf{r}_{j})\hat{\sigma}^{(j)}_{ge}$
(22)

propagating in the same direction. The underlying mechanism is based on the
parity symmetry of the atomic array, which interacts strongly with the
symmetric superposition, $\hat{a}^{(\rm in)}_{\rightarrow}+\hat{a}^{(\rm
in)}_{\leftarrow}$, and is completely decoupled from the anti-symmetric mode,
$\hat{a}^{(\rm in)}_{\rightarrow}-\hat{a}^{(\rm in)}_{\leftarrow}$. Yet,
misalignment of the optical elements may break the overall symmetry of the
setup and affect the ideal photon output Eq. (22). Hereby, however, the setup
can act as a sensitive interferometer to control and compensate such
imperfections, since any harmful misalignment will lead to a finite signal in
the second output field $\hat{b}$. For example a small displacement, $\Delta
z$, of the atomic array, generates a signal

$\displaystyle\langle\hat{b}^{\dagger}\hat{b}\rangle=4\pi^{2}\langle\hat{a}^{\dagger}\hat{a}\rangle\frac{\Delta
z^{2}}{\lambda^{2}},$ (23)

that can be employed for accurate phase control or adjustment of the beam
splitter arrangement.

To describe the interaction with photons at non-orthogonal incidence, we
consider linearly polarized light with the polarization vector oriented in the
plane of the atomic array irrespective of the angle $\theta$. Let us further
assume an additional magnetic field, oriented along the same direction, which
lifts the degeneracy between magnetic sub-levels and is tuned in such a way
that the incident field is near-resonant with a chosen $\pi$-transition. If
the Zeeman shifts of the atomic states exceed the atomic dipole-dipole
interaction, the magnetic field prevents the population of other excited
Zeeman states. This efficiently suppresses the coupling between different
polarization states of emitted photons and enables a simplified description in
terms of a single $|g\rangle-|e\rangle$ transition. Choosing a linear
polarization in the plane of the array, however, breaks rotational symmetry
and leads to an anisotropic interaction between the atoms as opposed to the
case of circular in-plane polarization. In addition, a finite incident angle
$\theta$ sets slightly more stringent constraints on the lattice constant

$\displaystyle a\leq\frac{\lambda}{1+\sin\theta}$ (24)

in order to suppress higher order Bragg-scattering into other modes and ensure
the collective suppression of photon losses.

Figure 6: Photon loss rate $\tilde{\gamma}$ in units of $\Gamma$ as a function
of the incident angle $\theta$ for different lattice constants $a$. Results
are shown for a rounded $31\times 31$ atomic array and a probe-beam waist of
$w_{0}=2\lambda$ to minimize finite-size effects for all considered values of
$a$. The applied laser fields resonantly ($\bar{\Delta}+\bar{\Delta}_{c}=0$)
excited $100S$ Rydberg states of Rb atoms with $\Omega/\Gamma=8.0$ and
$\Delta_{r}/\Gamma=-10.0$.

Photon losses out of the incident mode can be detected as a loss rate
$\tilde{\gamma}$, which generally yields a useful characterization of the
photon-coupling efficiency Solomons  _et al._ (2024). Here, we quantify photon
losses by comparing the transmission spectrum of the beam splitter setup with
the transmission

$\displaystyle
T(\delta)=\bigg{|}\frac{i(\tilde{\gamma}-\tilde{\Gamma})}{2\delta+i(\tilde{\gamma}+\tilde{\Gamma})}\bigg{|}^{2}$
(25)

of a single two-level emitter that is coupled to a single propagating photon
mode Sheremet  _et al._ (2023). By analogy with such an effective waveguide-
QED configuration and by fitting the corresponding expression Eq. (25) to the
calculated transmission spectrum, we can extract the photon loss rate
$\tilde{\gamma}$ and the effective emission rate $\tilde{\Gamma}$ into the
effective waveguide mode $\hat{a}$ (c.f. Fig.5). Figure 6 shows
$\tilde{\gamma}$ as a function of the incident angle $\theta$. Increasing
$\theta$ can significantly diminish the coupling efficiency due to the onset
of Bragg scattering into other modes. Such losses are suppressed for smaller
lattice constants, $a$, and vanish for $a<\lambda/2$, as discussed above [see
Eq.(24)]. In particular, one finds that experimentally achievable lattice
constants permit very low photon losses at sufficiently large angles to
implement the beam splitter setup of Fig.5. We can quantify its coupling
efficiency by the effective $\beta$-factor

$\displaystyle\beta=\frac{\tilde{\Gamma}}{\tilde{\Gamma}+\tilde{\gamma}},$
(26)

which is shown in Fig.7. Increasing the transverse beam waist $w_{0}$ reduces
imperfections due to finite transverse-momentum contributions ${\bf
k}_{\perp}$ Shahmoon  _et al._ (2017) of the incident mode
$u_{\rightarrow}({\bf r})$ and $u_{\leftarrow}({\bf r})$. Consequently, the
coupling efficiency $\beta$ increases with the beam waist, but eventually
saturates at a value $\beta_{\text{opt}}\lesssim 1$. Importantly, one can
reach this optimal value already for rather small beam waists, $w_{0}$, which
lie well below the typical Rydberg blockade radius for (cf. Fig.3).

Under these conditions, the asymptotic value

$\beta_{\text{opt}}=\frac{\bar{\Gamma}_{c}}{\bar{\Gamma}_{c}+\gamma}$ (27)

is only limited by the decay rate, $\gamma$, of the Rydberg state. It
therefore increases rapidly with the principal quantum number $n$, as
illustrated in Fig.7(b). Since the value of $\beta\sim 1-\gamma/\Gamma_{c}$ is
ultimately limited by the ratio of the Rydberg-state decay rate $\gamma$ and
the collective decay rate $\Gamma_{c}$ of $|e\rangle$, it can in principle be
further increased by working on the superradiative reflection resonance of the
atomic mirror that appears for smaller lattice constants $a\sim 0.2\lambda$.

Figure 7: (a) The coupling efficiency $\beta$ is shown as a function of the
probe-beam waist $w_{0}$ for different principal quantum numbers $n$. Panel
(b) shows the optimal $\beta_{\text{opt}}$ as a function of the Rydberg state
$n$. The calculations have been performed for a rounded $75\times 75$ array
with a lattice constant $a=0.65\lambda$ that is illuminated by a Gaussian beam
at an angle $\theta=5^{\circ}$. In panel (b) we have chosen a large beam waist
$w_{0}=8\lambda$. The remaining parameters are $\Omega_{p}/\Gamma=0.001$,
$\Omega/\Gamma=8.0$, and $\Delta_{r}/\Gamma=-10.0$. Figure 8: Comparison of
the output single photon amplitude, $\phi(t)$ between the case of a single
2-level atom coupled to a single waveguide mode and the case of the extended
Rydberg array coupled to a free-space pulse with Gaussian transverse spatial
beam profile. The temporal Gaussian shape of the incident pulse is shown in
green. It has a beam waist of $w_{0}=2.1\lambda$, a pulse length of
$\tau=2.24/\tilde{\Gamma}$, and resonantly ($\bar{\Delta}+\bar{\Delta}_{c}=0$)
excites Rydberg states with $n=100$. The remaining parameters are
$\Omega/\Gamma=8.0$, $a=0.75\lambda$, and $\Delta_{r}/\Gamma=-10.0$.

##  VI Photon pulse dynamics

Thus far, we have focused on cw-fields and studied the optical response of the
atomic array for a fixed frequency of the incident probe photons. This is
readily generalized by considering the operators

$\displaystyle\hat{a}_{\phi}(t)$ $\displaystyle=\int{\rm d}{\bf r}\>u^{*}({\bf
r})\phi^{*}({\bf e}{\bf r}-ct)\hat{\mathcal{E}}({\bf r},t)$ (28)

for a spatiotemporal mode of finite-length photon pulses that propagate along
the unit vector ${\bf e}$ with a pulse envelope $\phi(t)$ and a transverse
beam profile $u({\bf r})$. The incident fields are then described by
annihilation operators

$\displaystyle\hat{a}^{({\rm in})}_{\rightarrow}(t)$
$\displaystyle=\sqrt{\ell}\int{\rm d}{\bf r}\>u^{*}_{\rightarrow}({\bf
r})\phi^{*}({\bf e}_{\rightarrow}{\bf r}-ct)\hat{\mathcal{E}}({\bf r},t)$
$\displaystyle\hat{a}^{({\rm in})}_{\leftarrow}(t)$
$\displaystyle=\sqrt{\ell}\int{\rm d}{\bf r}\>u^{*}_{\leftarrow}({\bf
r})\phi^{*}({\bf e}_{\leftarrow}{\bf r}-ct)\hat{\mathcal{E}}({\bf r},t)$ (29)

for forward and backward propagating pulses along the antiparallel directions,
${\bf e}_{\rightarrow}$ and ${\bf e}_{\leftarrow}$, respectively, using the
mode functions introduced in Sec. II. The dynamics of an incident single-
photon pulse can then be studied, by simulating the atomic lattice under weak
coherent driving and projecting the state of the outgoing light onto the
single-photon sector. In this case, one can solve the master equation
(11)-(13) of the atomic lattice by propagating a single quantum trajectory
Mølmer  _et al._ (1993), $|\Psi(t)\rangle$ that describes the state of the $N$
atoms. The amplitude of the transmitted photon after propagating through the
beam splitter setup can then be obtained from $\psi(t)=\langle
0|\hat{a}(t)|\Psi(t)\rangle$, where $|0\rangle$ denotes the initial state of
the array with all atoms in their ground state. Here, the field operator
$\hat{a}(t)$ for the output light that defines the photon density along the
propagation direction as

$\hat{a}(t)=a^{({\rm in})}\phi(ct)+i\frac{\sqrt{2}g}{c}\sum_{j}u^{*}({\bf
r}_{j})\hat{\sigma}_{ge}^{(j)}(t)$ (30)

only acts on the atoms and corresponds to Eq.(22), but for a time dependent
coherent-state input field with an amplitude $a^{({\rm
in})}=\langle\hat{a}^{({\rm in})}\rangle\ll 1$. Figure 8 shows the simulated
photon amplitude for a Gaussian input pulse
$\phi(z)=e^{-\frac{z^{2}}{2c^{2}\tau^{2}}}/(\pi c^{2}\tau^{2})^{1/4}$. For the
chosen duration the spectral pulse width is comparable to the collective
linewidth of the array ($\tilde{\Gamma}\tau\sim 1$), such that the photon
undergoes significant pulse distortion. The result is in excellent agreement
with the pulse transmission by a single two-level emitter in a one-dimensional
waveguide-QED setting under perfect-coupling conditions, $\beta=1$.

Figure 9: Comparison of the wave function of the two-photon bound state
generated by (a) the Rydberg array and (b) a waveguide-coupled 2-level quantum
emitter for the same parameters as in Fig. 8.

Similarly, we can study the interaction with an incident two-photon pulse by
simulating the atomic array under weak coherent driving ($a^{({\rm in})}\ll
1)$ and calculating the two-photon output from $\psi_{2}(t,t^{\prime})=\langle
0|\hat{a}(t)\hat{a}(t^{\prime})|\Psi\rangle$. Also here, the obtained two-
photon output shown in Fig.9 under perfect coupling conditions ($\beta\sim 1$)
follows closely the behavior of the effective waveguide system with a single
saturable emitter and $\beta=1$. In this case Shen and Fan (2007a, b), the
correlated two-photon amplitude

$\displaystyle\psi_{2}(t,t^{\prime})=\psi(t)\psi(t^{\prime})+\psi_{b}(t,t^{\prime})$
(31)

can be interpreted in terms of a photonic bound state $\psi_{b}(t,t^{\prime})$
that is exponentially localized with respect to the time delay
$|t-t^{\prime}|$ between the two transmitted photons. Using Eq.(31), we can
formally extract $\psi_{b}$ from the numerical two-photon amplitude and indeed
find good agreement with the known analytical form Shen and Fan (2007a, b).
For a more quantitative comparison of the dynamical nonlinear response, we
have calculated the overlap between the two-photon output from the ideal
waveguide-QED setting and the Rydberg-atom array, and find and infidelity of
less than $10^{-3}$ for all studied parameters and principal quantum numbers
$40\leq n\leq 100$.

##  VII Applications

The obtained nonlinearity and directional photon coupling suggest a range of
applications, tapping into the capabilities of waveguide-QED with strongly
coupled quantum emitters Sheremet  _et al._ (2023). Below, we discuss two
examples: the generation of single photons and two-photon gate operations.

###  VII.1 Single photon generation

The availability of tunable single-photon pulses is an essential prerequisite
for a wide range of applications. Our three-level Rydberg array suggests a
direct approach for the deterministic generation of single-photon pulses via
photon storage and retrieval protocols with coherent input light.

Figure 10: (a) Temporal dynamics of the population in the driven mode
$\mathcal{E}(\mathbf{r})$ for a $\pi$ pulse. (b) The corresponding storage and
retrieval infidelity as a function of the beam size $w_{0}$. Here we have
chosen a $21\times 21$ array with $a=0.75\lambda$. In panel (a), the two laser
pulses generate a single Rydberg excitation in the blockaded array with a
control Rabi frequency $\Omega/\Gamma=4$, and detuning $\Delta_{r}/\Gamma=-20$
on the Raman resonance, where $\bar{\Delta}+\bar{\Delta}_{c}$=0. The probe
pulse has a transverse width of $w_{0}=3.0\lambda$, and temporal square-pulse
shape to generate a two-photon $\pi$-pulse with a duration $\Gamma\tau=10$.
Panel (b) shows the storage and retrieval infidelity of a single photon for
these parameters but a varying beam waist $w_{0}$ of the probe pulse. Figure
11: (a) Schematic of the photon sorting scheme based on an interferometer
setup that contains two Rydberg-atom arrays. Panel (b) shows a circuit that
utilizes this nonlinear element in combination with further linear operations
to implement an NS gate. The achievable sorting fidelity and success
probability is shown in (c) as a function of principal quantum number of the
Rydberg state.

First, one generates a single de-localized Rydberg excitation by illuminating
the array with classical light and exploiting the interaction blockade which
prevents excitation of more than a single Rydberg atom. In practice, this can
be accomplished in different ways, e.g., by adiabatically chirped excitation
of the effective two-level system, by light-storage on single-photon resonance
under EIT conditions, or by applying a $\pi$-pulse to invert the Rydberg-
blockaded superatom.

Here, we focus on the latter strategy. Since we are working far off resonance
(cf. Fig.2), we can consider the effective 2-level limit. While this
suppresses the attainable single-atom Rabi frequency, the Rydberg-blockade
collectively enhances the Rabi frequency of the $N$-atom array by a factor of
$\sim\sqrt{N}$. One can, thus, consider a $\pi$-pulse that generates a single
Rydberg excitation on a time scale that is much shorter than the typical
lifetime time of the excitation. We can, therefore, write the $N$-atom state
as
$|\Psi\rangle=c_{0}(t)|0\rangle+\sum^{N}_{j=1}c_{j}(t)\hat{\sigma}^{(j)}_{rg}|0\rangle$,
whose amplitudes evolve as

$\displaystyle\dot{c}_{0}=$ $\displaystyle i\sum_{j}\bar{\Omega}_{j}c_{j}$
(32a) $\displaystyle\dot{c}_{j}=$ $\displaystyle
i(\bar{\Delta}+i\frac{\gamma}{2})c_{j}+i\bar{\Omega}c_{0}+i\sum_{k\neq
j}(\bar{J}_{kj}+\frac{i}{2}\bar{\Gamma}_{kj})c_{k}.$ (32b)

with the time dependent two-photon Rabi frequency
$\bar{\Omega}_{j}=\bar{g}u(\mathbf{r}_{j})\phi(\bm{e}\mathbf{r}_{j}-ct)$. As
shown in Fig.10(a), one can obtain a near unit population

$\displaystyle
P_{u}=|\sum_{j}u^{*}(\mathbf{r}_{j})c_{j}|^{2}/\sum_{j}|u(\mathbf{r}_{j})|^{2},$
(33)

of Rydberg excitations in the spatial atomic mode defined by the excitation
pulse. In the depicted example, the chosen pulse length is
$\tau=10\Gamma^{-1}$, which is sufficiently short to suppress decoherence from
spontaneous decay and corresponds to an experimentally achievable value of
$\tau=260$ns for Rb atoms. Note that the dipole-dipole interaction tends to
distort the spatial profile of the spin wave excitation with respect to the
incident light. This effect plays only a minor role for moderate beam waists
of a few optical wavelengths and can be further suppressed by use of a wider
excitation beam. Figure 10(b) demonstrates that high storage efficiencies with
$1-P_{u}\sim 10^{-4}$ are possible with typical parameters.

The generated Rydberg-spinwave excitation can then be retrieved efficiently,
via an adiabatic readout under EIT conditions, i.e., by turning up the control
field $\Omega(t)$ on the $|r\rangle-|e\rangle$ transition. As shown in Manzoni
_et al._ (2018), sub-wavelength arrays can yield extraordinarily high
retrieval efficiencies, as the photon losses can be exponentially suppressed
compared to optical memories based on disordered atomic ensembles. The
retrieval dynamics is described by the simple set of evolution equations

$\displaystyle\frac{de_{j}}{dt}=$ $\displaystyle
i\Delta_{e}e_{j}+i\Omega^{*}(t)c_{j}+i\sum_{k}(J_{kj}+\frac{i}{2}\Gamma_{kj})e_{k}$
(34) $\displaystyle\frac{dc_{j}}{dt}=$ $\displaystyle
i(\Delta_{e}+\Delta_{r})c_{j}+i\Omega(t)c_{j}$ (35)

for the amplitudes $e_{j}$ and $c_{j}$ of an $|e\rangle$-state and Rydberg-
state excitation of the $j$th atoms, respectively. The polarization field of
the array yields the emission into the incident mode $\mathcal{E}({\bf r})$
and defines the retrieval efficiency Manzoni  _et al._ (2018)

$\displaystyle\eta=\frac{g^{2}}{c}\sum_{jl}u^{*}(\mathbf{r}_{j})u(\mathbf{r}_{l})\int^{\infty}_{0}e_{j}(t)e^{*}_{l}(t){\rm
d}t.$ (36)

Also here, the dipole-dipole interaction slightly distorts the mode of the
emitted field. This effect is reduced by employing a larger beam waist. For
typical experimental parameters, we find large fidelities of $\sim 0.99$ for
generating single photons in a single and fully controllable spatio-temporal
mode. For a homogeneous control field $\Omega$, the transverse mode function
is perfectly matched to that of the incident field. In addition, the temporal
mode profile can be shaped upon read-out by the time dependence of
$\Omega(t)$. Since this read-out field operates on single-photon resonance,
the relevant line width is given by $\Gamma_{c}$. Hence $\Omega(t)$ can be
varied rapidly compared to timescales
$\bar{\Gamma}^{-1}_{c}\gg{\Gamma}^{-1}_{c}$ while fully maintaining
adiabaticity for the retrieval process. This in turn enables precise temporal
control of the generated single-photon pulse. The $\pi$-pulse excitation
followed by resonant retrieval thus yields an efficient source of single
photons with full spatio-temporal tunability, making them useful for
controlled photon interactions with the nonlinear beam splitter setup
described in section V.

###  VII.2 Photon sorting and quantum gates

The high degree of tunability can be used for applications of the Rydberg-
mediated nonlinearity, as we shall discuss in this section. When two atom
arrays are integrated into the described beam splitter configuration of Fig.
11(a), the system can be thought of a chiral waveguide QED system containing
two cascaded two-level emitters. It has been shown in Ref. Yang  _et al._
(2022) that such an emitter pair is capable of performing a nearly
deterministic separation of single- and two-photon components into orthogonal
modes of a given input temporal pulse mode $\phi(t)$ [see Fig. 11(a)], i.e.,

$c_{1}\hat{a}_{\phi}^{\dagger}|0\rangle+c_{2}\frac{(\hat{a}_{\phi}^{\dagger})^{2}}{\sqrt{2}}|0\rangle~{}\rightarrow~{}c_{1}\hat{a}_{\psi}^{\dagger}|0\rangle+c_{2}\frac{(\hat{a}_{\theta}^{\dagger})^{2}}{\sqrt{2}}|0\rangle,$
(37)

where $\hat{a}_{f}^{\dagger}=\int dtf(t)\hat{a}^{\dagger}(t)$ creates a single
photon in the temporal mode $f(t)$ [c.f. Eq.(28)], with $\psi(t)$ and
$\theta(t)$ being two orthogonal ones [$\int dt\psi^{*}(t)\theta(t)=0$]. The
intuitive understanding of the above process is that for a single-photon input
state, its dispersion will accumulate when passing through two emitters,
resulting in a distorted output wavefunction $\psi(t)$; while for a two-photon
input, an approximate time reversal process is established: the second emitter
disentangles the correlated output wavefunction from the first one and
generates two uncorrelated photons occupying an undistorted but time-reversed
pulse $\theta(t)\approx\phi(T-t)$, with $T$ the delay time induced by the
emitters.

Perfect sorting of one- and two-photon components permits implementation of a
deterministic nonlinear-sign (NS) gate Witthaut  _et al._ (2012); Ralph  _et
al._ (2015), which can be further used to make the all-optical KLM quantum
computing protocol Knill  _et al._ (2001) deterministic. An NS gate flips the
sign of the two-photon component, while leaving the vacuum and single-photon
parts unaltered, i.e.,

$c_{0}|0\rangle+c_{1}\hat{a}_{\phi}^{\dagger}|0\rangle+c_{2}\frac{(\hat{a}_{\phi}^{\dagger})^{2}}{\sqrt{2}}|0\rangle~{}\rightarrow~{}c_{0}|0\rangle+c_{1}\hat{a}_{\phi}^{\dagger}|0\rangle-
c_{2}\frac{(\hat{a}_{\phi}^{\dagger})^{2}}{\sqrt{2}}|0\rangle,$ (38)

and it can be realized by using the interferometer setup shown in Fig. 11(b).
First, the same nonlinear element as in Fig. 11(a) is used to separate the
single-mode input field into photon-number dependent orthogonal temporal
modes. In a second step, these distinct temporal modes are transferred to
spatially separated modes. Such a temporal-mode beam splitter can be
implemented via sum frequency generation Eckstein  _et al._ (2011); Ansari
_et al._ (2018). A linear phase shifter can then be placed in the two-photon
path to induce a sign-flip of the two-photon amplitude, as illustrated in
Figs. 11(b). Following another temporal-mode beam splitter and a linear time
reversal operation, the photon pulses are sent through the other, thus far
unused, ports of the Rydberg-array interferometer to restore the original
single-mode nature of the input pulses, however, with an added sign-flip on
the two-photon component. The unidirectionally of the nonlinear element makes
it possible to use the Rydberg arrays twice in the quantum circuit, provided
one ensures a sufficient pulse delay to avoid interference of photons from the
different input ports of the Rydberg beam splitter setup.

In such a configuration, the Rydberg-array photon sorter is the only (passive)
nonlinear device, and all remaining single-qubit operations, such as temporal-
mode beam splitting Eckstein  _et al._ (2011); Ansari  _et al._ (2018) and
forward/backward time reversal Chumak  _et al._ (2010); Sivan and Pendry
(2011); Minkov and Fan (2018), can be performed with high fidelity. Figure
11(c) shows the attainable sorting fidelity and the corresponding success
probability for different principal quantum numbers $n$ of the Rydberg state.
For a $15\times 15$ atom array with $n=100$, the optimal sorting fidelity can
reach $\mathcal{F}\gtrsim 0.9998$ with a high success probability of
$\mathcal{P}>0.99$ for the quantum gate operation discussed above. This high
value is obtained for an array with lattice spacing $a/\lambda=0.65$. However,
further improvements are possible for denser arrays since this would increase
the effective coupling strength $\tilde{\Gamma}$. With an even higher
$\mathcal{P}\sim 1$, it may then become feasible to implement deterministic
Bell-state measurements and nonlinear phase gate operations on large photonic
quantum networks.

##  VIII Discussions and Conclusions

In this article, we have analyzed an approach to deploying subwavelength
atomic arrays as nonlinear optical elements in photonic quantum circuits.
Here, one exploits the near lossless collective coupling of extended arrays to
a single free-space optical mode, while large nonlinearities are obtained from
the strong interactions between highly excited Rydberg states of the atoms.
Since the direct coupling to high-lying states features intrinsically small
transition dipole moments, the combination of both effects requires three- or
multi-level coupling schemes, which leaves different possibilities to
introduce atomic Rydberg states. While Rydberg-dressing schemes Moreno-
Cardoner  _et al._ (2021) limit the range and strength of generated
interactions Gil  _et al._ (2014); Zeiher  _et al._ (2016) and resonant
3-level excitation Zhang  _et al._ (2022); Walther  _et al._ (2022); Solomons
and Shahmoon (2023) reduces the achievable nonlinear response Zhang  _et al._
(2022), we find here that driving a Rydberg Raman resonance under large
single-photon detunings offers optimal conditions for realising strong
effective photon interactions. This configuration yields a narrow reflection
resonance at which the nonlinear effect of Rydberg-state interactions is
maximized and can turn the array into a saturable 2-level quantum emitter with
near-perfect coupling to a free-space photonic mode.

This mode-insensitive response and the extended geometry of the array enable
the design of interferometric setups that behave like nonlinear unidirectional
elements. We illustrate potential applications, by showing that such elements
permit the deterministic generation of highly tunable single-photon pulses and
photon number sorting operations with high fidelities.

For two counterpropagating photons, the proposed element mediates a V-type
three-level non-linearity, which may be used to implement a controlled phase
gate Schrinski  _et al._ (2022). The expected fidelities enabled by the strong
collective light-matter coupling, thus, give promising perspectives for
implementations of deterministic gates for photonic quantum information
processing Graham  _et al._ (2022); Scholl  _et al._ (2023); Singh  _et al._
(2023); Evered  _et al._ (2023); Bluvstein  _et al._ (2024); Shah  _et al._
(2024) and quantum simulation Nishad  _et al._ (2023); Chen  _et al._ (2023);
Shaw  _et al._ (2024). Effective multi-emitter waveguide QED interactions
could further enable the preparation of useful non-Gaussian states of light
for quantum communication, metrology and sensing Eckner  _et al._ (2023);
Bornet  _et al._ (2023); Baßler  _et al._ (2024); Schäffner  _et al._ (2024);
Kleinbeck  _et al._ (2023); Lund  _et al._ (2024).

Cascading multiple arrays, the system may be used to explore many-body
dynamics in chiral waveguide QED, such as the propagation of multiphoton bound
states Mahmoodian  _et al._ (2020), repulsive photons Iversen and Pohl (2021,
2022), and superradiance bursts Liedl  _et al._ (2024). Hereby, adjusting the
transmittance of the beam splitters permits continuous tuning of the
chirality, which may facilitate studies of spontaneous symmetry breaking in
superradiance processes Cardenas-Lopez  _et al._ (2023).

Utilizing the Rydberg blockade in atomic arrays has enabled remarkable recent
breakthroughs in quantum computing Graham  _et al._ (2022); Scholl  _et al._
(2023); Singh  _et al._ (2023); Evered  _et al._ (2023); Bluvstein  _et al._
(2024); Shah  _et al._ (2024), quantum simulation Nishad  _et al._ (2023);
Chen  _et al._ (2023); Shaw  _et al._ (2024), quantum metrology and sensing
Eckner  _et al._ (2023); Bornet  _et al._ (2023); Baßler  _et al._ (2024);
Schäffner  _et al._ (2024) using atomic qubits, and the combination of the
Rydberg-states and optical dipole interactions offers broad perspectives for
exploring such applications with photonic quantum states.

###### Acknowledgements.

This work was supported by the Carlsberg Foundation through the “Semper
Ardens” Research Project QCooL, by the Danish National Research Foundation
through the Center of Excellence for Hybrid Quantum Networks (Grant agreement
No. DNRF139) and the Center of Excellence “CCQ” (Grant agreement No.:
DNRF152), by the Austrian Science Fund (Grant No. 10.55776/COE1) and the
European Union (NextGenerationEU), and by the Horizon Europe ERC Synergy Grant
SuperWave (Grant No. 101071882).

##  Appendix A Two-level approximation

To derive the effective two-level description in the limit of
$|\Delta_{e}|\gg|J_{ij}|,|\Gamma_{ij}|$, we start from the Heisenberg equation
for the transition operator

$\displaystyle\frac{d\hat{\sigma}^{(j)}_{ge}}{dt}=$ $\displaystyle
ig\hat{\mathcal{E}}(\mathbf{r}_{j})+i\big{(}\Delta_{e}+i\frac{\Gamma}{2}\big{)}\hat{\sigma}^{(j)}_{ge}+i\Omega^{*}\hat{\sigma}^{(j)}_{gr}$
$\displaystyle+i\sum_{m,j\neq
m}[J_{mj}+\frac{i}{2}\Gamma_{mj}]\hat{\sigma}^{(m)}_{ge}$ (39)

where $\hat{\sigma}^{(j)}_{gg}\approx\hat{I}$ in the the weak-driving limit
and we have omitted Langevin noise terms since we are exclusively considering
observables of normal ordered operators. For large $\Delta_{e}$, the equation
is readily solved perturbatively and yields to leading order in $1/\Delta_{e}$

$\displaystyle\hat{\sigma}^{(j)}_{ge}\simeq-\frac{g\hat{\mathcal{E}}(\mathbf{r}_{j})+\Omega^{*}\hat{\sigma}^{(j)}_{gr}}{\Delta_{e}}.$
(40)

Equation (II) then becomes

$\displaystyle\hat{H}_{0}\simeq-\sum_{j}\bar{\Delta}\hat{\sigma}^{(j)}_{rr}+[\bar{g}\hat{\mathcal{E}}(\mathbf{r}_{j})\hat{\sigma}^{(j)}_{rg}+h.c.]$
(41)

with the effective atom-light coupling strength $\bar{g}=-\Omega
g/\Delta_{e}$. Similarly, the dipole-dipole interaction can be rewritten as

$\displaystyle\hat{H}_{\text{dd}}$ $\displaystyle\simeq\sum_{j,k\neq
j}\frac{J_{jk}}{\Delta^{2}_{e}}[g\mathcal{E}^{*}(\mathbf{r}_{j})+\Omega\hat{\sigma}^{(j)}_{rg}][g\mathcal{E}(\mathbf{r}_{k})+\Omega^{*}\hat{\sigma}^{(k)}_{gr}]$
(42) $\displaystyle\simeq\sum_{j,k\neq
j}\bar{J}_{jk}\hat{\sigma}^{(j)}_{rg}\hat{\sigma}^{(k)}_{gr}$
$\displaystyle\mathcal{L}(\rho)$
$\displaystyle\simeq\sum_{i,j}\frac{\bar{\Gamma}_{ij}+\gamma\delta_{ij}}{2}(2\hat{\sigma}^{(i)}_{gr})\rho\hat{\sigma}^{(j)}_{rg}-\\{\hat{\sigma}^{(j)}_{rg}\hat{\sigma}^{(j)}_{gr}),\rho\\})$

where we have substituted Eq.(40) and
$\bar{J}_{jk}+i\bar{\Gamma}_{jk}/2=|\Omega|^{2}(J_{jk}+i\Gamma_{jk}/2)/\Delta^{2}_{e}$.
Throughout this work, assume coherent input fields for which we can replace
$\hat{a}^{(\text{in})}_{\rightarrow}$ and $\hat{a}^{(\text{in})}_{\leftarrow}$
by $a^{(\text{in})}_{\rightarrow}$ and $a^{(\text{in})}_{\leftarrow}$.
Substituting Eq.(40) into the input-output relations (43) then gives

$\displaystyle\hat{a}_{\rightarrow}(t)=$
$\displaystyle\hat{a}^{(\text{in})}_{\rightarrow}(t)+i\frac{\bar{g}}{c}\sqrt{\ell}\sum_{j}u^{*}(\mathbf{r}_{j})\hat{\sigma}^{(j)}_{gr}+i\varepsilon,$
(43a) $\displaystyle\hat{a}_{\leftarrow}(t)=$
$\displaystyle\hat{a}^{(\text{in})}_{\leftarrow}(t)+i\frac{\bar{g}}{c}\sqrt{\ell}\sum_{j}u^{*}(\mathbf{r}_{j})\hat{\sigma}^{(j)}_{gr}+i\varepsilon,$
(43b)

where
$\varepsilon=-\frac{g^{2}(a^{(\text{in})}_{\rightarrow}+a^{(\text{in})}_{\leftarrow})}{c\Delta_{e}}\ell\sum_{j}|u(\mathbf{r}_{j})|^{2}$
is a negligibly small correction to the total field that stems from the weak
photon emission from the far off-resonantly excited intermediate states.
Omitting this term, yields the input output relations (14) quoted in the main
text.

## References

  * Slussarenko and Pryde (2019) S. Slussarenko and G. J. Pryde, Applied Physics Reviews 6, 041303 (2019).
  * Bartlett  _et al._ (2021) B. Bartlett, A. Dutt, and S. Fan, Optica 8, 1515 (2021).
  * Aspuru-Guzik and Walther (2012) A. Aspuru-Guzik and P. Walther, Nature Physics 8, 285 (2012).
  * Hartmann (2016) M. J. Hartmann, Journal of Optics 18, 104005 (2016).
  * Giovannetti  _et al._ (2011) V. Giovannetti, S. Lloyd, and L. Maccone, Nature Photonics 5, 222 (2011).
  * Polino  _et al._ (2020) E. Polino, M. Valeri, N. Spagnolo, and F. Sciarrino, AVS Quantum Science 2, 024703 (2020).
  * Chang  _et al._ (2014) D. E. Chang, V. Vuletić, and M. D. Lukin, Nature Photonics 8, 685 (2014).
  * Birnbaum  _et al._ (2005) K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, Nature 436, 87 (2005).
  * Reiserer and Rempe (2015) A. Reiserer and G. Rempe, Rev. Mod. Phys. 87, 1379 (2015).
  * Welte  _et al._ (2018) S. Welte, B. Hacker, S. Daiss, S. Ritter, and G. Rempe, Phys. Rev. X 8, 011018 (2018).
  * Tiecke  _et al._ (2014) T. G. Tiecke, J. D. Thompson, N. P. de Leon, L. R. Liu, V. Vuletić, and M. D. Lukin, Nature 508, 241 (2014).
  * Petersen  _et al._ (2014) J. Petersen, J. Volz, and A. Rauschenbeutel, Science 346, 67 (2014).
  * Lodahl  _et al._ (2015) P. Lodahl, S. Mahmoodian, and S. Stobbe, Rev. Mod. Phys. 87, 347 (2015).
  * Chang  _et al._ (2018) D. E. Chang, J. S. Douglas, A. González-Tudela, C.-L. Hung, and H. J. Kimble, Rev. Mod. Phys. 90, 031002 (2018).
  * Yu  _et al._ (2019) S.-P. Yu, J. A. Muniz, C.-L. Hung, and H. J. Kimble, Proceedings of the National Academy of Sciences 116, 12743 (2019).
  * Prasad  _et al._ (2020) A. S. Prasad, J. Hinney, S. Mahmoodian, K. Hammerer, S. Rind, P. Schneeweiss, A. S. Sørensen, J. Volz, and A. Rauschenbeutel, Nature Photonics 14, 719 (2020).
  * Peyronel  _et al._ (2012) T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, Nature 488, 57 (2012).
  * Paris-Mandoki  _et al._ (2017a) A. Paris-Mandoki, C. Braun, J. Kumlin, C. Tresp, I. Mirgorodskiy, F. Christaller, H. P. Büchler, and S. Hofferberth, Phys. Rev. X 7, 041010 (2017a).
  * Murray and Pohl (2016) C. Murray and T. Pohl, Adv. At. Mol. Opt. Phys. 65, 321 (2016).
  * Firstenberg  _et al._ (2016) O. Firstenberg, C. S. Adams, and S. Hofferberth, Journal of Physics B: Atomic, Molecular and Optical Physics 49, 152003 (2016).
  * Thompson  _et al._ (2017) J. D. Thompson, T. L. Nicholson, Q.-Y. Liang, S. H. Cantu, A. V. Venkatramani, S. Choi, I. A. Fedorov, D. Viscor, T. Pohl, M. D. Lukin, and V. Vuletić, Nature 542, 206 (2017).
  * Yang  _et al._ (2020) F. Yang, Y.-C. Liu, and L. You, Phys. Rev. Lett. 125, 143601 (2020).
  * Facchinetti  _et al._ (2016) G. Facchinetti, S. D. Jenkins, and J. Ruostekoski, Phys. Rev. Lett. 117, 243601 (2016).
  * Bettles  _et al._ (2016) R. J. Bettles, S. A. Gardiner, and C. S. Adams, Phys. Rev. Lett. 116, 103602 (2016).
  * Shahmoon  _et al._ (2017) E. Shahmoon, D. S. Wild, M. D. Lukin, and S. F. Yelin, Phys. Rev. Lett. 118, 113601 (2017).
  * Perczel  _et al._ (2017) J. Perczel, J. Borregaard, D. E. Chang, H. Pichler, S. F. Yelin, P. Zoller, and M. D. Lukin, Phys. Rev. Lett. 119, 023603 (2017).
  * Guimond  _et al._ (2019) P.-O. Guimond, A. Grankin, D. V. Vasilyev, B. Vermersch, and P. Zoller, Phys. Rev. Lett. 122, 093601 (2019).
  * Ballantine and Ruostekoski (2020) K. E. Ballantine and J. Ruostekoski, Phys. Rev. Lett. 125, 143604 (2020).
  * Grankin  _et al._ (2018) A. Grankin, P. O. Guimond, D. V. Vasilyev, B. Vermersch, and P. Zoller, Phys. Rev. A 98, 043825 (2018).
  * Baßler  _et al._ (2023) N. S. Baßler, M. Reitz, K. P. Schmidt, and C. Genes, Opt. Express 31, 6003 (2023).
  * Pedersen  _et al._ (2023) S. P. Pedersen, L. Zhang, and T. Pohl, Phys. Rev. Res. 5, L012047 (2023).
  * Bekenstein  _et al._ (2020) R. Bekenstein, I. Pikovski, H. Pichler, E. Shahmoon, S. F. Yelin, and M. D. Lukin, Nature Physics 16, 676 (2020).
  * Moreno-Cardoner  _et al._ (2021) M. Moreno-Cardoner, D. Goncalves, and D. E. Chang, Phys. Rev. Lett. 127, 263602 (2021).
  * Zhang  _et al._ (2022) L. Zhang, V. Walther, K. Mølmer, and T. Pohl, Quantum 6, 674 (2022).
  * Sheremet  _et al._ (2023) A. S. Sheremet, M. I. Petrov, I. V. Iorsh, A. V. Poshakinskiy, and A. N. Poddubny, Rev. Mod. Phys. 95, 015002 (2023).
  * Gross and Haroche (1982) M. Gross and S. Haroche, Physics Reports 93, 301 (1982).
  * Dung  _et al._ (2002) H. T. Dung, L. Knöll, and D.-G. Welsch, Phys. Rev. A 66, 063810 (2002).
  * Ficek and Tanaś (2002) Z. Ficek and R. Tanaś, Physics Reports 372, 369 (2002).
  * Manzoni  _et al._ (2018) M. T. Manzoni, M. Moreno-Cardoner, A. Asenjo-Garcia, J. V. Porto, A. V. Gorshkov, and D. E. Chang, New Journal of Physics 20, 083048 (2018).
  * Fleischhauer  _et al._ (2005) M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
  * Rui  _et al._ (2020) J. Rui, D. Wei, A. Rubio-Abadal, S. Hollerith, J. Zeiher, D. M. Stamper-Kurn, C. Gross, and I. Bloch, Nature 583, 369 (2020).
  * Browaeys and Lahaye (2020) A. Browaeys and T. Lahaye, Nature Physics 16, 132 (2020).
  * Ebadi  _et al._ (2021) S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. W. Ho, S. Choi, S. Sachdev, M. Greiner, V. Vuletić, and M. D. Lukin, Nature 595, 227 (2021).
  * Scholl  _et al._ (2021) P. Scholl, M. Schuler, H. J. Williams, A. A. Eberharter, D. Barredo, K.-N. Schymik, V. Lienhard, L.-P. Henry, T. C. Lang, T. Lahaye, A. M. Läuchli, and A. Browaeys, Nature 595, 233 (2021).
  * Semeghini  _et al._ (2021) G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kalinowski, R. Samajdar, A. Omran, S. Sachdev, A. Vishwanath, M. Greiner, V. Vuletić, and M. D. Lukin, Science 374, 1242 (2021).
  * Satzinger and et. al. (2021) K. J. Satzinger and et. al., Science 374, 1237 (2021).
  * Saffman  _et al._ (2010) M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod. Phys. 82, 2313 (2010).
  * Lukin  _et al._ (2001) M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 87, 037901 (2001).
  * Jaksch  _et al._ (2000) D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, Phys. Rev. Lett. 85, 2208 (2000).
  * Wu  _et al._ (2021) X. Wu, X. Liang, Y. Tian, F. Yang, C. Chen, Y.-C. Liu, M. K. Tey, and L. You, Chinese Physics B 30, 020305 (2021).
  * Dudin and Kuzmich (2012) Y. O. Dudin and A. Kuzmich, Science 336, 887 (2012).
  * Paris-Mandoki  _et al._ (2017b) A. Paris-Mandoki, C. Braun, J. Kumlin, C. Tresp, I. Mirgorodskiy, F. Christaller, H. P. Büchler, and S. Hofferberth, Phys. Rev. X 7, 041010 (2017b).
  * Ripka  _et al._ (2018) F. Ripka, H. Kübler, R. Löw, and T. Pfau, Science 362, 446 (2018).
  * Li  _et al._ (2003) W. Li, I. Mourachko, M. W. Noel, and T. F. Gallagher, Phys. Rev. A 67, 052502 (2003).
  * Beterov  _et al._ (2009) I. I. Beterov, I. I. Ryabtsev, D. B. Tretyakov, and V. M. Entin, Phys. Rev. A 79, 052504 (2009).
  * Roy  _et al._ (2017) D. Roy, C. M. Wilson, and O. Firstenberg, Rev. Mod. Phys. 89, 021001 (2017).
  * Thompson  _et al._ (2013) J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletić, and M. D. Lukin, Science 340, 1202 (2013).
  * Arcari  _et al._ (2014) M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, Phys. Rev. Lett. 113, 093603 (2014).
  * Tiranov  _et al._ (2023) A. Tiranov, V. Angelopoulou, C. J. van Diepen, B. Schrinski, O. A. D. Sandberg, Y. Wang, L. Midolo, S. Scholz, A. D. Wieck, A. Ludwig, A. S. Sørensen, and P. Lodahl, Science 379, 389 (2023).
  * Kannan  _et al._ (2020) B. Kannan, M. J. Ruckriegel, D. L. Campbell, A. Frisk Kockum, J. Braumüller, D. K. Kim, M. Kjaergaard, P. Krantz, A. Melville, B. M. Niedzielski, A. Vepsäläinen, R. Winik, J. L. Yoder, F. Nori, T. P. Orlando, S. Gustavsson, and W. D. Oliver, Nature 583, 775 (2020).
  * Solomons  _et al._ (2024) Y. Solomons, R. Ben-Maimon, and E. Shahmoon, PRX Quantum 5, 020329 (2024).
  * Mølmer  _et al._ (1993) K. Mølmer, Y. Castin, and J. Dalibard, J. Opt. Soc. Am. B 10, 524 (1993).
  * Shen and Fan (2007a) J.-T. Shen and S. Fan, Phys. Rev. A 76, 062709 (2007a).
  * Shen and Fan (2007b) J.-T. Shen and S. Fan, Phys. Rev. Lett. 98, 153003 (2007b).
  * Yang  _et al._ (2022) F. Yang, M. M. Lund, T. Pohl, P. Lodahl, and K. Mølmer, Phys. Rev. Lett. 128, 213603 (2022).
  * Witthaut  _et al._ (2012) D. Witthaut, M. D. Lukin, and A. S. Sørensen, Europhys. Lett. 97, 50007 (2012).
  * Ralph  _et al._ (2015) T. Ralph, I. Söllner, S. Mahmoodian, A. White, and P. Lodahl, Phys. Rev. Lett. 114, 173603 (2015).
  * Knill  _et al._ (2001) E. Knill, L. Raymond, and J. M. Gerald, Nature 409, 46 (2001).
  * Eckstein  _et al._ (2011) A. Eckstein, B. Brecht, and C. Silberhorn, Opt. Express 19, 13770 (2011).
  * Ansari  _et al._ (2018) V. Ansari, J. M. Donohue, M. Allgaier, L. Sansoni, B. Brecht, J. Roslund, N. Treps, G. Harder, and C. Silberhorn, Phys. Rev. Lett. 120, 213601 (2018).
  * Chumak  _et al._ (2010) A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, Nat. Commun. 1, 141 (2010).
  * Sivan and Pendry (2011) Y. Sivan and J. B. Pendry, Phys. Rev. Lett. 106, 193902 (2011).
  * Minkov and Fan (2018) M. Minkov and S. Fan, Phys. Rev. B 97, 060301 (2018).
  * Gil  _et al._ (2014) L. I. R. Gil, R. Mukherjee, E. M. Bridge, M. P. A. Jones, and T. Pohl, Phys. Rev. Lett. 112, 103601 (2014).
  * Zeiher  _et al._ (2016) J. Zeiher, R. van Bijnen, P. Schauß, S. Hild, J.-y. Choi, T. Pohl, I. Bloch, and C. Gross, Nature Physics 12, 1095 (2016).
  * Walther  _et al._ (2022) V. Walther, L. Zhang, S. F. Yelin, and T. Pohl, Phys. Rev. B 105, 075307 (2022).
  * Solomons and Shahmoon (2023) Y. Solomons and E. Shahmoon, Phys. Rev. A 107, 033709 (2023).
  * Schrinski  _et al._ (2022) B. Schrinski, M. Lamaison, and A. S. Sørensen, Phys. Rev. Lett. 129, 130502 (2022).
  * Graham  _et al._ (2022) T. M. Graham, Y. Song, J. Scott, C. Poole, L. Phuttitarn, K. Jooya, P. Eichler, X. Jiang, A. Marra, B. Grinkemeyer, M. Kwon, M. Ebert, J. Cherek, M. T. Lichtman, M. Gillette, J. Gilbert, D. Bowman, T. Ballance, C. Campbell, E. D. Dahl, O. Crawford, N. S. Blunt, B. Rogers, T. Noel, and M. Saffman, Nature 604, 457 (2022).
  * Scholl  _et al._ (2023) P. Scholl, A. L. Shaw, R. B.-S. Tsai, R. Finkelstein, J. Choi, and M. Endres, Nature 622, 273 (2023).
  * Singh  _et al._ (2023) K. Singh, C. E. Bradley, S. Anand, V. Ramesh, R. White, and H. Bernien, Science 380, 1265 (2023).
  * Evered  _et al._ (2023) S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara, H. Levine, G. Semeghini, M. Greiner, V. Vuletić, and M. D. Lukin, Nature 622, 268 (2023).
  * Bluvstein  _et al._ (2024) D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, J. P. Bonilla Ataides, N. Maskara, I. Cong, X. Gao, P. Sales Rodriguez, T. Karolyshyn, G. Semeghini, M. J. Gullans, M. Greiner, V. Vuletić, and M. D. Lukin, Nature 626, 58 (2024).
  * Shah  _et al._ (2024) F. Shah, T. L. Patti, O. Rubies-Bigorda, and S. F. Yelin, Phys. Rev. A 109, 012613 (2024).
  * Nishad  _et al._ (2023) N. Nishad, A. Keselman, T. Lahaye, A. Browaeys, and S. Tsesses, Phys. Rev. A 108, 053318 (2023).
  * Chen  _et al._ (2023) C. Chen, G. Bornet, M. Bintz, G. Emperauger, L. Leclerc, V. S. Liu, P. Scholl, D. Barredo, J. Hauschild, S. Chatterjee, M. Schuler, A. M. Läuchli, M. P. Zaletel, T. Lahaye, N. Y. Yao, and A. Browaeys, Nature 616, 691 (2023).
  * Shaw  _et al._ (2024) A. L. Shaw, Z. Chen, J. Choi, D. K. Mark, P. Scholl, R. Finkelstein, A. Elben, S. Choi, and M. Endres, Nature 628, 71 (2024).
  * Eckner  _et al._ (2023) W. J. Eckner, N. Darkwah Oppong, A. Cao, A. W. Young, W. R. Milner, J. M. Robinson, J. Ye, and A. M. Kaufman, Nature 621, 734 (2023).
  * Bornet  _et al._ (2023) G. Bornet, G. Emperauger, C. Chen, B. Ye, M. Block, M. Bintz, J. A. Boyd, D. Barredo, T. Comparin, F. Mezzacapo, T. Roscilde, T. Lahaye, N. Y. Yao, and A. Browaeys, Nature 621, 728 (2023).
  * Baßler  _et al._ (2024) N. S. Baßler, A. Aiello, K. P. Schmidt, C. Genes, and M. Reitz, Phys. Rev. Lett. 132, 043602 (2024).
  * Schäffner  _et al._ (2024) D. Schäffner, T. Schreiber, F. Lenz, M. Schlosser, and G. Birkl, PRX Quantum 5, 010311 (2024).
  * Kleinbeck  _et al._ (2023) K. Kleinbeck, H. Busche, N. Stiesdal, S. Hofferberth, K. Mølmer, and H. P. Büchler, Phys. Rev. A 107, 013717 (2023).
  * Lund  _et al._ (2024) M. M. Lund, F. Yang, V. R. Christiansen, D. Kornovan, and K. Mølmer, arXiv:2404.12328 (2024).
  * Mahmoodian  _et al._ (2020) S. Mahmoodian, G. Calajó, D. E. Chang, K. Hammerer, and A. S. Sørensen, Phys. Rev. X 10, 031011 (2020).
  * Iversen and Pohl (2021) O. A. Iversen and T. Pohl, Phys. Rev. Lett. 126, 083605 (2021).
  * Iversen and Pohl (2022) O. A. Iversen and T. Pohl, Phys. Rev. Res. 4, 023002 (2022).
  * Liedl  _et al._ (2024) C. Liedl, F. Tebbenjohanns, C. Bach, S. Pucher, A. Rauschenbeutel, and P. Schneeweiss, Phys. Rev. X 14, 011020 (2024).
  * Cardenas-Lopez  _et al._ (2023) S. Cardenas-Lopez, S. J. Masson, Z. Zager, and A. Asenjo-Garcia, Phys. Rev. Lett. 131, 033605 (2023).