## Abstract

Nonlinear optical phenomena are typically local. Here, we predict the possibility of highly nonlocal optical nonlinearities for light propagating in atomic media trapped near a nano-waveguide, where long-range interactions between the atoms can be tailored. When the atoms are in an electromagnetically induced transparency configuration, the atomic interactions are translated to long-range interactions between photons and thus to highly nonlocal optical nonlinearities. We derive and analyze the governing nonlinear propagation equation, finding a roton-like excitation spectrum for light and the emergence of order in its output intensity. These predictions open the door to studies of unexplored wave dynamics and many-body physics with highly nonlocal interactions of optical fields in one dimension.

© 2016 Optical Society of America

## 1. INTRODUCTION

Optical nonlinearities are commonly described by local nonlinear responses
of the material to the optical field, resulting in the dependence of the
refractive index at point $z$
on the field at the same point, $E(z)$ [1]. Recently however, there has been a
growing interest in nonlocal nonlinear optics, namely, in mechanisms
whereby the refractive index at $z$
depends on the field intensity at different points
${z}^{\prime}$
in the material [2]. Mechanisms that
give rise to such nonlocal nonlinearities include heat diffusion [3,4], molecular reorientation in liquid crystals [5], and atomic diffusion [6,7]. This paper discusses a new regime of *extremely
nonlocal* nonlinearities affecting both the frequency and the
*quadratic dispersion* of optical waves [see Eq. (1) below]. The physical
mechanism that leads to this new regime is very different than those
explored previously [2–7]: it relies on an atomic medium
prepared in an electromagnetically induced transparency (EIT)
configuration whose optical nonlinearity is *controlled* by
shaping the nonlocal *dipolar interactions* between the
atoms.

EIT is associated with the lossless and slow propagation of light pulses in resonant atomic media subject to the coherent driving of an auxiliary atomic transition [8,9]. Since the early days of EIT, it has been explored as a means of enhancing optical nonlinearities [8,10–12]. A particularly effective mechanism for giant optical nonlinearities is provided by dipolar interactions between atoms that form the medium. Since EIT can be described by the propagation of the so-called dark-state polariton [13], which is a superposition of the light field and an atomic spin wave, the inherently nonlocal dipolar interactions between atoms are translated to nonlocal nonlinearities in polariton propagation. In the case of dipolar interactions between Rydberg atoms in free space, most theoretical [14–22] and experimental [23–27] studies have focused on their remarkable strength, a useful feature in quantum information. Exploration of the nonlocal aspect of this nonlinearity, on the other hand, has received less attention, and was restricted to the nonlocality dictated by the $1/{r}^{6}$ spatial dependence of the van der Waals potential between Rydberg atoms [28].

The present work rests on two recently explored mechanisms: that of EIT
polaritons and that of *modified long-range dipolar
interactions* in confining geometries, such as fibers,
waveguides, photonic band structures, and transmission lines, which are
currently attracting considerable interest [29–38]. Yet, we show that the combined effect of these mechanisms may
allow for new and unfamiliar possibilities of nonlocal nonlinear optics.
More specifically, we show how dispersive laser-induced dipolar
interactions between atoms coupled to a nano-waveguide with a grating,
which can be designed to extend over *hundreds* of optical
wavelengths [32,33], are translated via EIT into
extremely nonlocal optical nonlinearities. We then analyze light
propagation in this medium along the waveguide and find a roton-like
excitation spectrum for light and the emergence of spatial self-order in
its output intensity.

## 2. SUMMARY AND SCOPE OF THE RESULTS

The main general result of this work is the derivation of a nonlinear
propagation equation for the (possibly quantum) EIT polariton field
$\widehat{\mathrm{\Psi}}$
comprised of light *guided* by the waveguide, and an atomic
medium *tightly* trapped along the waveguide axis
$z$
with an effectively one-dimensional (1D) inter-atomic potential
$U(z-{z}^{\prime})$,

*nonlocal*nonlinear detuning ${\widehat{\delta}}_{\mathrm{NL}}$, which depends on field intensities integrated over the medium with the interaction kernel $U(z)$. The appearance of a

*nonlocal nonlinearity*not only in the frequency shift but also in the

*dispersion coefficient*gives rise to a new regime of nonlocal nonlinear optics. The physical system and reasoning that lead to Eq. (1), as well as its derivation, are discussed in Section 3 below. In principle, we assume sufficient conditions for a lossless EIT propagation (coefficients $v$, $\alpha $, and $C$ are real), whereas loss and decoherence mechanisms due to imperfections and scattering are analyzed in Section 6.

The second part of this work (Sections 4 and 5) is dedicated to study some first implications of wave propagation in such a medium, and specifically to the analysis of wave excitations on top of a continuous-wave (CW) background. General results for the excitation spectrum (dispersion relation) and the intensity correlations at the output are presented in Section 4, Eqs. (8) and (11), respectively. These are followed in Section 5 by specific results for a medium of atoms trapped near a waveguide grating that supports extremely long-range interactions $U(z)$, as per Eq. (12). The results for the excitation spectrum exhibit a roton-like narrow-band shape, which may be probed by a homodyne detection scheme (Fig. 3). The roton-like spectrum signifies the tendency of light in this regime to exhibit spatial self-order, namely, crystal-like correlations; these can be revealed by measuring the photon intensity at the waveguide’s output (Fig. 4).

The predicted self-order of light constitutes a new, hitherto unexplored, optical “phase,” analogous to the spatial structure of cold atomic media subject to light-induced dipolar interactions [32,39–44]. We discuss important aspects and prospects of this work in Section 7.

## 3. EIT POLARITONS WITH NONLOCAL INTERACTIONS

#### A. System

Consider a medium of identical atoms in an EIT configuration as in Fig. 1: the atoms are trapped at a distance ${r}_{a}$ from a nano-waveguide along its longitudinal $z$-axis [31,45–54] [Fig. 1(a)]. A strong (external) coupling field with a constant and uniform Rabi frequency $\mathrm{\Omega}$ drives the $|d\u27e9\to |e\u27e9$ atomic transition with detuning ${\delta}_{c}$ and wavenumber ${k}_{c}$, whereas a weak (possibly quantized) probe field with carrier frequency ${\omega}_{0}$, wavenumber ${k}_{0}$, and envelope $\widehat{\mathcal{E}}$ is resonantly coupled to the $|g\u27e9\to |e\u27e9$ transition (${\omega}_{0}={\omega}_{eg}$) [Fig. 1(b)]. Under tight transverse trapping (around ${r}_{a}$) with respect to the transition wavelength, the atomic positions can be characterized by their longitudinal component $z$ [see Supplement 1, Section 1.A] [32,45]. We assume the existence of a dipolar interaction $U(z)$ between atoms that occupy the state $|d\u27e9$ [Fig. 1(c) and Section 5.A below]. Then, the energy of level $|d\u27e9$ of an atom at $z$ is shifted by ${\delta}_{\mathrm{NL}}\sim {n}_{a}\int \mathrm{d}{z}^{\prime}U(z-{z}^{\prime}){P}_{d}({z}^{\prime})$, ${n}_{a}$ being the atomic density (per unit length) and ${P}_{d}({z}^{\prime})$ the occupation of state $|d\u27e9$ in an atom at ${z}^{\prime}$.

We may now explain the physical reasoning that leads to Eq. (1). In Fig. 2, we plot the complex linear susceptibility $\chi $ of the EIT medium to the probe field $\widehat{\mathcal{E}}$ as a function of its detuning ${\mathrm{\Delta}}_{p}$ in the presence of a coupling field detuned by ${\mathrm{\Delta}}_{c}$ [8], which in our case is given by ${\mathrm{\Delta}}_{c}={\delta}_{c}+{\delta}_{\mathrm{NL}}$ [Fig. 1(b)]. When ${\mathrm{\Delta}}_{c}=0$ [Fig. 2(a)], the absorption coefficient $\mathrm{Im}\text{\hspace{0.17em}}\chi $ is symmetric with respect to ${\mathrm{\Delta}}_{p}$, whereas the dispersion $\mathrm{Re}\text{\hspace{0.17em}}\chi $ is antisymmetric, so no (real) quadratic dispersion exists for the probe envelope centered around ${\mathrm{\Delta}}_{p}=0$. By contrast, when ${\mathrm{\Delta}}_{c}\ne 0$ is introduced [Fig. 2(b)], $\mathrm{Re}\text{\hspace{0.17em}}\chi $ is no longer antisymmetric and quadratic dispersion exists, which explains the term ${\mathrm{\Delta}}_{c}C{v}^{2}{\partial}_{z}^{2}$ in Eq. (1). However, this comes at the price of non-vanishing losses at ${\mathrm{\Delta}}_{p}=0$ (see also Section 6). For this reason, we choose to work in the so-called Autler–Townes regime, $\mathrm{\Omega}\gg \gamma $, ${\mathrm{\Delta}}_{c}$ [8], where $\gamma $ is the width of the level $|e\u27e9$. Then, for ${\mathrm{\Delta}}_{c}$ smaller than the single-atom transparency window, ${\mathrm{\Delta}}_{c}\ll {\mathrm{\Omega}}^{2}/\gamma ,\mathrm{\Omega}$, but still larger than $\gamma $, the absorption per atom can become negligible while dispersion is still significant, as illustrated in Fig. 2(b) (see also Supplement 1, Section 1). This explains the lossless propagation described by Eq. (1) with real parameters $\alpha $, $v$, $C$. As long as the absorption, associated with dissipation due to spontaneous emission at rate $\gamma $, is negligible, so are the noise effects of vacuum fluctuations; Eq. (1) then holds in operator form without additional Langevin quantum noise operators.

#### B. Derivation of Eq. (1)

The formal derivation of Eq. (1) goes as follows (see Supplement 1, Section 1 for more details). The field envelope $\widehat{\mathcal{E}}(z)=\sum _{k}{\widehat{a}}_{k}{e}^{ikz}/\sqrt{L}$, with commutation relations $[{\widehat{a}}_{k},{\widehat{a}}_{k}{\prime}^{\u2020}]={\delta}_{k{k}^{\prime}}$ and hence $[\widehat{\mathcal{E}}(z),{\widehat{\mathcal{E}}}^{\u2020}({z}^{\prime})]=\delta (z-{z}^{\prime})$, is assumed to be spectrally narrow and is guided by a transverse mode of the waveguide (later taken to be the ${\mathrm{HE}}_{11}$ mode of a fiber) with effective area $A$ at the atomic position ${r}_{a}$ and polarization vector ${\mathbf{e}}_{0}$. The Hamiltonian in the interaction picture is

Moving to the polariton picture of EIT [13], we define the dark and bright polaritons, $\widehat{\mathrm{\Psi}}$ and $\widehat{\mathrm{\Phi}}$, respectively,

In the adiabatic regime, where the probe field is a CW and in the absence of detunings (${\delta}_{c}$, $U=0$), the bright polariton $\widehat{\mathrm{\Phi}}$ vanishes [13]. Here, we take the first non-adiabatic correction [Supplement 1, Section 1.C] by inserting the equation for $\widehat{\mathrm{\Phi}}$ into that of $\widehat{\mathrm{\Psi}}$, and we assume all detunings to be smaller than the EIT transparency window ${\delta}_{\mathit{tr}}={\mathrm{\Omega}}^{2}/(\gamma \sqrt{OD})$, with $OD=({n}_{a}/A)L{\sigma}_{a}$ and ${\sigma}_{a}$ the cross section of the $|g\u27e9\to |e\u27e9$ transition, finally arriving at Eq. (1) with $\alpha ={\mathrm{sin}}^{2}\text{\hspace{0.17em}}\theta $, $v=c\text{\hspace{0.17em}}{\mathrm{cos}}^{2}\text{\hspace{0.17em}}\theta $, and $C={\mathrm{sin}}^{2}\text{\hspace{0.17em}}\theta (2-3\text{\hspace{0.17em}}{\mathrm{sin}}^{2}\text{\hspace{0.17em}}\theta )/{|\mathrm{\Omega}|}^{2}$.

## 4. COLLECTIVE EXCITATIONS AND PROPAGATION IN A CW BACKGROUND

#### A. Excitation Spectrum of Polariton Waves

With Eq. (1) in hand, we now turn to the analysis of the polariton wave propagation it predicts. Specifically, we consider the CW polariton solution and find the dispersion relation of wave excitations around this CW background, analogous to the Bogoliubov spectrum of excitations in a Bose–Einstein condensate (BEC) [55,56]. The CW solution of Eq. (1) is $\psi (t)={\psi}_{0}{e}^{-i(\alpha {\delta}_{c}+{n}_{p}{U}_{0})t}$, with ${\psi}_{0}=|{\psi}_{0}|{e}^{i\phi}$, ${n}_{p}={\alpha}^{2}{|{\psi}_{0}|}^{2}$ being an effective polariton density per unit length and ${U}_{0}$ the $k=0$ component of the spatial Fourier transform of the potential ${U}_{k}={\int}_{-\infty}^{\infty}\mathrm{d}zU(z){e}^{-ikz}$. Here we have neglected edge effects by assuming $l<z<L-l$, $l$ being the range of the potential $U(z)$. The dispersion relation of small fluctuations $\varphi (z,t)$ around the large average CW field $\u27e8\mathrm{\Psi}\u27e9=\psi (t)$ are found as usual upon inserting $\mathrm{\Psi}=\psi (t)+\varphi (z,t)$ into Eq. (1) and linearizing it by keeping the fluctuations $\varphi $ to linear order. Then, introducing the ansatz [55],

This means that a polariton wave distortion (about the CW solution) with a wavenumber $k$ relative to the carrier wavenumber (inside the EIT medium) oscillates at a frequency (relative to the carrier frequency ${\omega}_{0}$)

where the $\pm $ sign is for positive/negative $k$, respectively. The dispersion relation [Eq. (9)] is composed of the detuning $\alpha {\delta}_{c}$ due to the coupling field, the self-phase modulation of the CW component ${n}_{p}{U}_{0}$ (analogous to the chemical potential in a BEC [55]), the linear dispersion $vk$, and the modified Bogoliubov excitation spectrum ${\omega}_{k}$. The spectrum ${\omega}_{k}$ is determined by the interplay between the interaction Fourier transform ${U}_{k}$ and the “kinetic-energy” quadratic dispersion ${\omega}_{k}^{0}$, which is affected by both the detuning ${\delta}_{c}$ and by the $k=0$ component of ${U}_{k}$. This interplay is further discussed below for $U(z)$ resulting from laser-induced interactions near a waveguide grating.#### B. Generation of Two-Mode Correlations

The parametric process described by the foregoing modified Bogoliubov theory also entails the dynamic generation of two-mode squeezing, i.e., pairs of entangled polaritons with wavenumbers $\pm k$. The analysis is similar to that of propagation in fibers with local Kerr nonlinearity [57,58]. Upon inserting the expansion of small quantum fluctuations in the longitudinal wavenumber modes $k$, $\widehat{\varphi}(z)=\sum _{k}{\widehat{a}}_{k}{e}^{ikz}/\sqrt{L}$, into the linearized equation for $\varphi (z,t)$, we obtain coupled first-order differential equations (in time) for ${\widehat{a}}_{k}(t)$ and ${\widehat{a}}_{-k}^{\u2020}(t)$, whose solution is a dynamic Bogoliubov transformation [Supplement 1, Section 2.B],

## 5. HIGHLY NONLOCAL LASER-INDUCED INTERACTION VIA WAVEGUIDE GRATING

Our analysis up to this point was kept general, without specifying the interaction potential $U(z)$. Let us now turn to a particularly interesting case of an extremely long-range interaction, where novel nonlinear optical effects can be illustrated.

#### A. Shaping the Interaction Potential

The illumination of atoms by an off-resonant laser virtually excites the atoms and allows them to resonantly interact via virtual photons. The spatial dependence of the resulting interaction potential $U(z)$ then follows the spatial structure of the mediating photon modes. This is the essence of the laser-induced interaction potential we wish to employ [59]. Specifically, consider another laser with Rabi frequency ${\mathrm{\Omega}}_{L}$, which is detuned by $|{\delta}_{L}|\gg {\mathrm{\Omega}}_{L}$ from the transition $|d\u27e9\to |s\u27e9$, $|s\u27e9$ being a fourth atomic level [Fig. 1(c)], and assume that this transition is distinct and separated from the transitions used for EIT [Fig. 1(b)], either spectrally or by polarization. Then, the extended waveguide modes can mediate long-range interactions between the trapped atoms [Fig. 1(a)] [32]. As a specific example, we consider a nano-waveguide that incorporates a grating, i.e., periodic perturbation of the refractive index with period $\mathrm{\Lambda}\equiv \pi /{k}_{B}$, so that the photon modes exhibit a bandgap (see, e.g., Refs. [48,60]). Then, for a laser frequency ${\omega}_{L}$ inside the gap (the probe field’s carrier frequency ${\omega}_{0}$ being outside the gap), the laser-induced interaction potential becomes [32] (Supplement 1, Section 3)

The resulting spatial Fourier transform ${U}_{k}$ then consists of four Lorentzian peaks of width $\sim 1/l$ centered around the spatial beating frequencies $\pm ({k}_{L}^{z}-{k}_{B})$ and $\pm ({k}_{L}^{z}+{k}_{B})$. We note that the laser ${\mathrm{\Omega}}_{L}$ and the grating are unrelated to the linear propagation of the probe field and their sole role is to induce the long-range dipolar interaction Eq. (12) between atoms, which is in turn translated via EIT to interaction between polaritons as in Eq. (1) (see also Discussion, Section 7.B).

#### B. Roton and Anti-Roton Spectra

Let us focus on the peak of ${U}_{k}$ around ${k}_{R}\equiv {k}_{L}^{z}-{k}_{B}$ and its effect on the dispersion relation (spectrum), shown in Eq. (8). We first consider the case of anomalous dispersion, where the signs of ${\omega}_{k}^{0}$ and ${U}_{k}$ are opposite. Analogous to BEC, this describes the case of an attractive potential ${U}_{k}$ that competes with the “kinetic energy” ${\omega}_{k}^{0}$. Then, for $k$-values satisfying $|{\omega}_{k}^{0}|>2{n}_{p}|{U}_{k}|$, ${\omega}_{k}$ is real and exhibits a dip around ${k}_{R}$, in contrast to the case of a local potential, for which ${U}_{k}$ is independent of $k$ (standard Bogoliubov spectrum) and this feature is absent. This is seen in Fig. 3(a), where both the analytical results of Eq. (8) and numerical simulations of the nonlinear Eq. (1) (Supplement 1, Section 5) are plotted and shown to agree very well. The narrow-band “dip” of this ${\omega}_{k}$ spectrum is in analogy with the roton minimum in He II [61]. It reflects the fact that wave distortions about the CW field with spatial frequencies around ${k}_{R}$ cost less energy and are hence favorable. This feature implies that the intensity of the polariton field in its ground state would tend to self-order with typical wavenumber ${k}_{R}$ [61].

Turning to the case of normal dispersion, where the signs of ${\omega}_{k}^{0}$ and ${U}_{k}$ are identical, ${\omega}_{k}$ exhibits an “anti-roton” peak around ${k}_{R}$ [Fig. 3(b)]. This means that distortions of spatial frequencies around ${k}_{R}$ are costly, so the system prefers to avoid these spatial variations. This behavior again manifests the tendency of the system to order, since it indicates the spatial distortions that the system is unlikely to be found in should it be in its ground state.

In order to measure the roton and anti-roton spectra, we first recall the meaning of the dispersion relation $\omega (k)$ from Eq. (9): without an interaction, the frequency associated with a wave envelope at wavenumber $k$ traveling inside the EIT medium is given by ${\omega}^{(0)}(k)=\alpha {\delta}_{c}+vk+{\delta}_{c}C{v}^{2}{k}^{2}=\omega (k)-{n}_{p}{U}_{0}-({\omega}_{k}-{\delta}_{c}C{v}^{2}{k}^{2})$, so that ${\omega}_{k}$ (together with ${n}_{p}{U}_{0}$) expresses a frequency, or phase velocity, shift due to the nonlinearity. Namely, the wavenumber $k$ describes a spatial eigenmode of propagation, both with and without interaction, with eigenfrequencies $\omega (k)$ and ${\omega}^{(0)}(k)$, respectively. Now, suppose we let a weak quasi-CW pulse of length ${L}_{p}<L$ and frequency ${\omega}_{p}={\omega}^{(0)}(k)$ enter the medium (on top of the strong CW) when the laser and hence the interaction ${U}_{k}$ are turned off. Since upon entering the medium, the field does not change its frequency ${\omega}_{p}$, we deduce from the dispersion relation in the absence of interaction ${\omega}^{(0)}(k)$ that the field inside the medium exhibits a perturbation $\varphi (z)$ at spatial frequency $k$ on top of the strong CW. Subsequently, when the entire pulse is in the medium, we immediately (non-adiabatically) turn on the laser ${\mathrm{\Omega}}_{L}$ and hence the interaction ${U}_{k}$, so that the temporal frequency of the perturbation $\varphi (z)$ at wavenumber $k$ becomes $\omega (k)={\omega}^{(0)}(k)+{n}_{p}{U}_{0}+({\omega}_{k}-{\delta}_{c}C{v}^{2}{k}^{2})$. The frequency shift ${n}_{p}{U}_{0}+({\omega}_{k}-{\delta}_{c}C{v}^{2}{k}^{2})$ can be thought of as an extra energy acquired by the mode $k$ due to the interaction energy. Therefore, the spectrum ${\omega}_{k}$ can be inferred from the frequency shift, measurable by homodyne detection of the pulse $\varphi (z)$ that exits the EIT medium [Fig. 3(c)]. A similar procedure was proposed for measuring the tachyon-like spectrum of polaritons in inverted media [62].

#### C. Dynamical Instability: Pair Generation

In the anomalous dispersion case, consider now a sufficiently strong interaction such that for a narrow band of $k$-values around ${k}_{R}$, where ${U}_{k}$ is peaked, the condition $2{n}_{p}|{U}_{k}|>|{\omega}_{k}^{0}|$ is satisfied and ${\omega}_{k}$ becomes imaginary. Then, field perturbations around ${k}_{R}$ become exponentially unstable [Fig. 4(a)], resulting in parametric amplification and the generation of entangled photon pairs in this narrow band of unstable $k$-values. The strength of the amplified perturbation and generated field is characterized by the magnitude of the coefficients ${\mu}_{k}(t)$ and ${\nu}_{k}(t)$ from Eq. (10), which grow exponentially with propagation time $t$ and are largest for the narrow peak around ${k}_{R}$. The resulting squeezing spectrum ${G}_{k}$ at the output $t=L/v$ [Fig. 4(a)] then exhibits stronger squeezing at a narrow bandwidth around ${k}_{R}$, which may be measured by homodyne detection [57].

#### D. Dynamical Instability: Emergence of Self-Order

An interesting implication of the extremely nonlocal potential of
Eq. (12) is the
*dynamic formation of order* in the system.
Consider that at $t=0$,
there exist fluctuations around the CW, with a spatial spectrum
${N}_{k}={N}_{0}{e}^{-{(k/{q}_{\mathrm{tr}})}^{2}}$,
i.e., “$\delta $-correlated”
noise limited by the EIT transparency window of width
${\delta}_{\mathrm{tr}}=v{q}_{\mathrm{tr}}$.
Then, fluctuations at $k$-values
around the peak ${k}_{R}$
will be parametrically amplified as they propagate through the medium,
as verified in Fig. 4(b), where
the spatial spectrum of the polariton field, ${N}_{k}(t)=\u27e8{\widehat{a}}_{k}^{\u2020}(t){\widehat{a}}_{k}(t)\u27e9$,
at different propagation times $t$
($t=L/v$
describing the output field) is calculated both analytically and via
direct (classical) numerical simulations of Eq. (1). This suggests that the
system becomes spatially ordered with a period
$\sim 2\pi /{k}_{R}$,
which may be revealed by measuring the correlation function
${g}^{(2)}$
between the intensities of the output field that arrive at a detector
at the waveguide’s end ($z=L$)
and time difference $(z-{z}^{\prime})/v$
[scheme from Fig. 3(c) without
the lower local oscillator arm]. We obtain the corresponding
${g}^{(2)}$
by numerically integrating over $k$
in the classical limit of Eq. (11), where the vacuum-fluctuation contributions are
neglected. This calculation is compared to ${g}^{(2)}$
measured via direct numerical simulations of Eq. (1), yielding excellent
agreement. Figure 4(c) reveals
the emergence of order by presenting the intensity correlations
${g}^{(2)}$
at different propagation times $t$
through the medium ($t=L/v$
at the output). The resulting ${g}^{(2)}$
exhibits oscillations with a period $z-{z}^{\prime}\sim 2\pi /({k}_{R})$, which persist
over a few $l$.
Considering that the interaction range $l$
can reach hundreds or even thousands of optical wavelengths
($l\approx 3027{\lambda}_{L}$
and $L\sim 10l\sim 0.026\text{\hspace{0.17em}}\mathrm{m}$ in our example,
see Supplement
1, Section 4), the light intensity
clearly becomes ordered due to the long-range interaction
$U(z)$, as
can also be seen by the comparison to the local interaction case
[inset of Fig. 4(c)].

## 6. SCATTERING AND IMPERFECTIONS

So far, we have considered a purely coherent evolution of the polariton field. Here we address three main sources of scattering and loss of field excitations; we estimate the decoherence rate each of them imposes on the polariton field and its possible effect on the observability of the self-ordering effects discussed above (see Supplement 1, Section 6, for details).

First, since the interaction $U(z)$ from Fig. 1(b) is induced by the illumination of the atoms by an off-resonant laser, it is accompanied by an incoherent process of scattering of laser photons ${\mathrm{\Omega}}_{L}$ from the $|d\u27e9\to |s\u27e9$ transition to non-guided modes at rate ${R}_{fs}$ [59]. This process limits the coherence time of the ${\overline{\sigma}}_{gd}$ spin wave and hence that of the polariton to be below $\sim {R}_{fs}^{-1}$, which in the examples of Figs. 3 and 4 is nevertheless much longer than the experiment time $L/v$ (Supplement 1, Section 4).

Second, consider material imperfections in the waveguide grating structure (e.g., defects and surface roughness) that give rise to the scattering of photons off the guided modes. This leads to a decay rate (width) $\kappa $ for each of the longitudinal (guided) photon modes $k$ (see Supplement 1 Section 6.C and [33,49]). Then, since atoms at level $|d\u27e9$ are coupled via ${\mathrm{\Omega}}_{L}$ to these lossy waveguide grating modes, the ${\overline{\sigma}}_{gd}$ spin wave and hence the polariton are decohered at a rate ${R}_{im}^{(1)}\sim {U}_{L}/\mathrm{Im}\sqrt{1-i\kappa /(2{\delta}_{u})}$, ${\delta}_{u}$ being the detuning of the laser ${\mathrm{\Omega}}_{L}$ from the upper band edge of the grating. However, the effect of ${R}_{im}^{(1)}$ on the spectra ${\omega}_{k}$ and the observables mentioned above can in principle be made arbitrary small by noting that ${\omega}_{k}$ depends on ${n}_{p}{U}_{L}$, whereas ${R}_{im}^{(1)}$ depends solely on ${U}_{L}$, reflecting the fact that ${R}_{im}^{(1)}$ is a single-polariton loss mechanism, whereas ${\omega}_{k}$ is a cooperative effect. Then, decreasing $|{U}_{L}|$ while keeping ${n}_{p}{U}_{L}$ constant (by increasing the CW power ${n}_{p}$) reduces ${R}_{im}^{(1)}$ but keeps ${\omega}_{k}$ unchanged.

Nevertheless, a cooperative decoherence can in fact result from $\kappa $ represented by an imaginary part acquired by $U(z)$; namely, a pair of atoms at a distance $z$ apart can jointly scatter photons at a rate $\sim \mathrm{Im}\text{\hspace{0.17em}}U(z)$, since the virtual photons that mediate their interaction $U$ are now lossy and enable excitation decay to non-guided modes. Then, for a single polariton at $z$, the scattering induced by the entire atomic medium becomes ${R}_{im}^{(2)}\sim {n}_{p}\int \mathrm{d}{z}^{\prime}\text{\hspace{0.17em}}\mathrm{Im}\text{\hspace{0.17em}}U({z}^{\prime})\sim {n}_{p}|{U}_{0}|\kappa {\mathrm{\Delta}}_{u}/({\kappa}^{2}/4+{\mathrm{\Delta}}_{u}^{2})$, with ${\mathrm{\Delta}}_{u}\approx {\omega}_{L}{({k}_{R}/{k}_{B})}^{2}/(\overline{n}\mathrm{\Delta}n)$. In Supplement 1, Section 6.B, we show that in cases where this dephasing channel limits the observability of ${\omega}_{k},{\gamma}_{k}$, its effect can be reduced by increasing the EIT coupling laser $\mathrm{\Omega}$.

Finally, we recall that EIT losses were neglected considering all detunings and the probe bandwidth to be within the transparency width ${\delta}_{\mathit{tr}}$. The EIT losses analysis of Supplement 1, Section 6.C goes beyond these simple considerations and treats the EIT-induced decoherence rate ${R}_{\mathrm{EIT}}$ using the full EIT loss spectrum. We find that the observability of the anti-roton peak is indeed not affected much by the EIT decoherence, whereas a similar conclusion is reached in the case of the instability by possibly detuning the carrier frequency of the probe field to match that of the coupling field (two-photon resonance). EIT losses may, however, pose a significant limitation to the observability of the more sensitive roton “dip.”

In principle, all loss terms discussed above should be accompanied by corresponding quantum noise terms, which are not fully considered here, restricting this discussion to the classical field case (for which all of the above results apply, excluding the squeezing spectrum ${G}_{k}$).

## 7. DISCUSSION

This study predicts a new and hitherto unexplored regime of nonlinear optics; namely, that of highly nonlocal interactions between photons in 1D. These nonlocal optical nonlinearities arise for light propagation inside driven atomic media in the vicinity of a waveguide and affect both the frequency and quadratic dispersion of the light field. We have derived the nonlinear equation that governs light propagation in this regime and have analyzed it around its CW solution, finding a narrow roton-like dispersion relation [Figs. 3(a) and 3(b)] and squeezing (entanglement) spectrum [Fig. 4(a)] and the emergence of order in the field intensity [Figs. 4(b) and 4(c)], all of which reflect the tendency of the system to self-organize, which in turn results from the long-range interactions between photons. Tremendous experimental progress in both key ingredients of our scheme has been recently reported and constantly being pursued, namely, that of electromagnetically induced transparency in waveguide systems [51,63] and the ability to design dipolar interactions therein [64]. In the following, we wish to discuss some important aspects of this work.

#### A. Structure and Generality of Eq. (1)

It is important to note the *nonlocal and nonlinear
dispersion* term appearing in Eq. (1), absent in previous
works on nonlocal nonlinear optics [2,4–7] and EIT-based nonlinear optics
[14–18,20,22,28]. We
identify the physical conditions under which this term becomes
significant and may lead to new phenomena, namely, in the
Autler–Townes regime of EIT and for sufficiently large nonlinear
detuning.

Another interesting point concerns the generality of this work in
relation to *different confining geometries*. The
derivation of Eq. (1)
does not depend on the specific form of the potential
$U(z)$,
which naturally opens the way to the exploration of nonlocal nonlinear
optics due to laser-induced dipolar potentials mediated by confined
photon modes of geometries other than the waveguide grating considered
here. For example, the laser-induced potential in a cavity along
$z$
is in general not translational-invariant and depends on the
$z$-positions
of both interacting atoms (rather than just their difference), a case
which is qualitatively different from the one analyzed here.

A different possibility is to consider modified laser-induced dipolar
interactions shaped by *controlling the spectrum of the
exciting laser*
${\mathrm{\Omega}}_{L}$,
rather than by working in a confined geometry [65,66].
Using our approach, this may allow exploring the generalization of
Eq. (1) to a
three-dimensional (free-space) environment with a designed dipolar
potential.

#### B. Origin and Length Scale of Self-Ordering

We stress that the length scale associated with order ${k}_{R}={k}_{L}\text{\hspace{0.17em}}\mathrm{cos}{\theta}_{L}-{k}_{B}$ originates from the interaction potential of the light field with itself, Eq. (12); hence, ordering spontaneously occurs in this optical system, much like in other condensed-mater systems and crystals. This is in contrast to, e.g., the order of atoms in a deep-potential optical lattice, where the atoms are situated at lattice sites determined by the potential imposed by an external laser rather than by their mutual interactions. In our case, the role of the grating is not to trap the propagating polariton, but merely to create dispersive and long-range dipolar interactions between atoms $U(z)$ (induced by an external laser ${\mathrm{\Omega}}_{L}$ that is unrelated to the light component of the polariton), which underlies the nonlinearity in the polariton propagation.

Moreover, it is important to note that the length scale
${k}_{R}$
is a signature of the specific spatial shape of the potential
$U(z)$
[sinusoidal in the case of Eq. (12)], as also revealed by the excitation and
instability spectra of Figs. 3
and 4(a). This is in contrast
to, e.g., the polariton crystallization process in a
*short-range* potential described in Ref. [19] using Luttinger’s
liquid theory, where the specific shape of the potential is
irrelevant. In this respect, our results are more related to the
modulational instability discussed in Ref. [28] for Rydberg-atom EIT, though there the
spatial dependence of the potential is not tunable and is restricted
to a power law (no length scale). However, the considered system is
three-dimensional and the quadratic dispersion is linear and emerges
simply due to diffraction. Other recent works where photon spatial
correlations follow those of the interaction potential include
Refs. [24] and [67,68], which, however, treat a probe field with
only a few photons (typically two), in contrast with the largely
collective behavior discussed in the present work.

#### C. Prospects

To conclude, this work opens the way to experimental and theoretical
investigations of new nonlinear wave phenomena, especially by
venturing beyond the linearized regime and exploring the role of the
nonlocal and nonlinear dispersion term ${\widehat{\delta}}_{\mathrm{NL}}C{v}^{2}{\partial}_{z}^{2}$.
Specific directions of further research may include: *nonlocal
nonlinear optics in 1D*, concerning the study of solitons,
where the lensing effect created by the nonlinear refractive index
change is now highly nonlocal, following $U(z)$;
*thermalization in 1D*, which has been considered
both experimentally [69] and
theoretically [70] for isolated
BEC in 1D, may become qualitatively different here due to the nonlocal
character of the designed interactions $U(z)$ and the
nonlinear dispersion (“mass”); and *effects of
non-additivity* of systems with long-range interactions
[71] may be studied here
for quantum/classical optical fields.

## Funding

Austrian Science Fund (FWF) [P25329-N27, SFB F41 (VICOM), I830-N13 (LODIQUAS)]; Israel Science Foundation (ISF); United States–Israel Binational Science Foundation (BSF).

## Acknowledgment

We acknowledge discussions with Ofer Firstenberg, Arno Rauschenbeutel, Philipp Schneeweiß, Darrick Chang, Nir Davidson, Tommaso Caneva, and Boris Malomed.

See Supplement 1 for supporting content.

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