# Controlling radiative properties of circular atomic arrays through nanofiber guided modes

Juliette Jiménez Jaimes, S. Nic Chormaic, E. Brion

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Juliette Jiménez Jaimes, S. Nic Chormaic, E. Brion. Controlling radiative properties of circular atomic arrays through nanofiber guided modes. Physical Review A, 2025, 112 (5), pp.053702. 10.1103/91kf-xl7p. hal-05125702

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Submitted on 23 Jun 2025

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Distributed under a Creative Commons Attribution 4.0 International License
Collective emission of atomic nanorings around an optical nanofiber

J. Jiménez-Jaimes¹, S. Nic Chormaic², E. Brion¹

¹Laboratoire Collisions Agrégats Réactivité, UMR 5589, FeRMI,
Université de Toulouse, CNRS, 118 Route de Narbonne, F-31062 Toulouse, France,

²OIST Graduate University, Onna-son, Okinawa 904-0495, Japan

We theoretically investigate the collective emission of one and two circular arrays of two-level
atoms surrounding an optical nanofiber. We show that the radiation eigenmodes of a single ring
selectively couple to specific guided modes of the fiber, according to their symmetry, and study how
the physical parameters of the system control their nature. In particular, we identify situations
in which the emission toward radiation modes is highly suppressed with respect to fiber-guided
modes while the lifetime of the atomic excitation is enhanced. We further address the case of two
identical nanorings positioned at a distance from each other along the nanofiber. By contrast to free-
space configurations, the rings can exchange excitations even at large separation through nanofiber
guided modes resulting in enhanced sub- and supperradiance with respect to the single-ring case.
Our findings suggest that the ring configuration is promising for the implementation of efficient
and versatile light-matter nanofiber-based interfaces and the achievement of waveguide quantum
electrodynamics.

I. INTRODUCTION

Light-matter interactions are at the core of most pro-
posals and implementations in quantum technologies.
Ongoing progresses in nanofabrication make it possible
to design nanowaveguides that offer both strong light
confinement and tailored optical guided modes. Ta-
pered subwavelength optical fibers, also called optical
nanofibers (ONFs), are among the most versatile and
promising light-matter interfaces [1]. Relatively easy to
fabricate in the laboratory, ONFs possess a great wealth
of guided modes [2, 3] with an intense evanescent com-
ponent [4, 5]. Interaction with these modes is instru-
mental to detect and trap quantum emitters such as
atoms, molecules, quantum dots or NV centers in di-
amond around the ONF [6–15]. The generation, life-
time, interactions, and trapping of highly excited Ry-
dberg atoms in the vicinity of a silica ONF have also
been investigated, both theoretically and experimentally
[16–20]. Due to the interactions ONFs provide between
a single guided mode and a large number of surround-
ing atoms, nanofiber-based platforms also pave the way
to the implementation of waveguide quantum electro-
dynamics (WQED) [21] and the observation of collec-
tive phenomena such as super-, sub- or selective radi-
ance [22]. Superradiance was already demonstrated in
a system of two atomic clouds 0.3 mm apart from each
other and coupled via the evanescent field of an optical
nanofiber [23]. A collective excitation was also imprinted
then read out via the DLCZ protocol in an array of 133
Cs atoms trapped along an ONF [24]. EIT-based slow-
light and light storage were reported both in an atomic
cloud overlapping an ONF [25] and in a regular array of
atoms trapped in an optical lattice realized in the evanes-
cent field around the ONF [26]. Large Bragg reflection
was also demonstrated in an atomic array trapped in the
evanescent field of an ONF, with both unidirectional and
chiral interactions observed [27].

Subradiant states of ordered atomic arrays have been

particularly studied, theoretically [28–33] and experimen-
tally [34–38], for their potential applications to quan-
tum memories [22] and transport [39]. The control of
their lifetime was investigated in 1D arrays in free space
[22, 40–42], near a waveguide [43–48], as well as in 2D
arrays [49–51]. Because of its unique symmetry proper-
ties, the ring geometry was also considered in free-space
and shown to potentially lead to enhanced single-photon
sensing, transport, storage, and light generation in engi-
neered nanoscale systems [22, 30, 39, 52–58].

Herein we investigate the emission of two-level atoms
arranged in a regular nanoring configuration around
an ONF. We show that the different radiation eigen-
modes (REMs) of a singly excited ring selectively cou-
ple to specific nanofiber guided modes (NFGMs) depend-
ing on their symmetry properties and that, by properly
choosing the parameters and state of the system, one
can make the ring emit predominantly towards a given
NFGM while strongly suppressing losses towards radia-
tion modes (Sec.II). We also show that two nanorings
can exchange an excitation, even at a large separation
distance, via the NFGMs, and that the REMs of the two-
ring system resulting from this coupling exhibit enhanced
sub- and supperradiance with respect to the single-ring
case (Sec.III). Our study is not restricted to the single-
mode regime, contrary to most previous works on col-
lective radiance in atomic systems in the vicinity of an
ONF [59, 60]. Though many other configurations could
and will be considered in the future, this work already
demonstrates the great potential of interfacing atomic
nanorings with an ONF for implementing highly selective
and enhanced coupling to NFGMs and therefore entering
the realm of WQED.

II. A SINGLE RING

We start by considering the case of a single ring com-
prising *N* two-level atoms surrounding a silica ONF of
<figure><img src="image_2.png" /><figcaption>FIG. 1: (a,b) Circular array of N = 5 identical two-level atoms around an ONF of axis (Oz). ρ and a denote the radii of the array and the nanofiber, respectively. The position vector and polar angle of atom (j) are denoted by r_j and θ_j ≡ 2π(j-1)/N, respectively. Atomic dipoles, p_j, are either (a) orthoradial, p_j ≡ p e_θ (θ_j), or (b) along the z-axis, p_j ≡ p e_z. (c) Two identical nanorings, separated by the distance Δz.</figcaption></figure>

optical index n_fiber = 1.45 (see Fig. 1 for hypotheses and notations). Throughout this article, we fix N = 5 ; although this choice is arbitrary it is actually not critical since the typical features obtained in this case are also observed for 2 ≤ N ≤ 10, as verified numerically. When the atomic system is initially singly excited, it evolves according to the effective – symmetric but nonHermitian – Hamiltonian [22, 52]

$$H_{\text{eff}} = -\frac{3\pi\hbar\gamma_0}{k_0} \sum_{i,j=1}^{N} g_{ij} \sigma_+^{(i)} \sigma_-^{(j)} \quad (1)$$

where γ₀ is the single-atom spontaneous emission rate in free space, gᵢⱼ ≡ uᵢ\* · G(rᵢ, rⱼ, k₀) · uⱼ is the coupling between atoms i and j, uᵢ ≡ pᵢ/||pᵢ|| is the normalized transition dipole moment of atom i, k₀ ≡ ω₀/c and ω₀ are the wavevector and frequency of the atomic transition, respectively, σᵢ∓ is the ith atom raising/lowering operator, and G is the Maxwell Green’s dyadic tensor in the presence of the fiber [17, 61, 62]. Note that G splits into its free-space, G₀, guided, G₉, and radiation mode contributions, Gᵣ, and the same decomposition applies to gᵢⱼ’s. The eigenvectors and eigenvalues of Hₐ₇₅ are denoted by {|Ψₙ⟩, −3πħγ₀/k₀λₙ} with the characteristic index n conventionally taking the integer values n = [−N/2] · · · [N/2], where [·] denotes the ceiling value. Since Hₐ₇₅ is not Hermitian, the λₙ's are a priori complex and the |Ψₙ⟩'s are a priori not orthogonal to each other according to the Hermitian scalar product. These states are the ring radiation eigenmodes (REMs) with associated decay rates Γₙ ≡ −6πħγ₀/k₀Imλₙ. For a

regular nanoring, one has [52]

$$\lambda_n = \frac{1}{N} \sum_{kl} e^{-i n \frac{2\pi(k-l)}{N}} g_{kl} \quad (2)$$

$$|\Psi_n\rangle = \frac{1}{\sqrt{N}} \sum_j e^{i n \frac{2\pi(j-1)}{N}} |e_j\rangle \quad (3)$$

where |eⱼ⟩ denotes the state in which only the jth atom is excited.

In Fig. 2, we plot the decay rates of the REMs n = 0, 1, 2 as functions of λ₀/d – here d = 2ρsin(π/N) is the distance between two neighbouring atoms in the ring – (a) in free space, Γₙ⁽⁽⁾⁾⁾, and (b) in the presence of the fiber, Γₙ⁽⁽⁾⁾⁾, in the case of orthoradial dipoles (Fig. 1a). The two plots show similar peak structures, nevertheless, in the close vicinity of the ONF – here ρ = 1.1a, the strongly collective behavior range is shifted towards higher values of the ratio λ₀/d, which we attribute to the effect of the ONF optical index, while the superradiant peaks and subradiant dips are more pronounced.

In Fig. 2c, we plot the ratios Γₙ⁽⁽⁾⁾⁾/Γₙ⁽⁽⁾⁾⁾ of the decay rates from the REMs n = 0, 1, 2, towards the nanofiber guided modes (NFGMs), Γₙ⁽⁽⁾⁾⁾, over the total spontaneous emission rates of the same REMs, Γₙ⁽⁽⁾⁾⁾, as functions of the ratio λ₀/d. Figs. 2d-g further show the separate contributions of the different NFGMs which are found to couple only to specific REMs. In a mode-function approach [63], the coupling between the REM n and a NFGM Kᵢ≥₀,ₘ, with K = HE, EH, TE, TM, of polarization vector eᵢₘ^K and azimuthal number pl, with l ≥ 0 and p = ±, is indeed proportional to ∑ⱼ eᵢ²π/N(n+pl)(j-1)uⱼ · eᵢₘ^K(rⱼ). For orthoradial atomic dipoles, uⱼ = u_θ(rⱼ), this term simplifies to ∑ⱼ eᵢ²π/N(n+pl)(j-1) [eᵢₘ^K(rⱼ)]_θ. Since [eᵢₘ^TM(rⱼ)]_θ = 0 for all j = 1 · · · N, NFGMs TM₀ₘ do not couple to any REM as confirmed by Fig. 2f. Moreover, since [eᵢₘ^K=HE,EH,TE(rⱼ)]_θ does not depend on the polar angle θ [63], the coupling of REMs to NFGMs Kᵢₘ is proportional to ∑ⱼ=1...N eᵢ²π/N(n+pl)(j-1) = N × δᵢₙₚl[...] where δᵢₙₚl[...] is nonzero only for n + pl = kN with k ∈ Z. Therefore NFGMs TE₀₁, HE₁₁, HE₂₁ – and more generally TE₀ₘ, HE₁ₘ, HE₂ₘ – couple to REMs n = 0, ±1, ±2, respectively (Figs. 2d,e,g). In particular, at λ₀/d ≈ 3.4, the REM n = 1 is slightly subradiant, Γₙ⁽⁽⁾⁾⁾/γ₀ ≈ 0.6 (see Fig. 2b), because the contribution of free-space and radiative modes is strongly suppressed hence maximizing the probability for the spontaneously emitted photon to be captured by the NFGM HE₁₁, Γₙ⁽⁽⁾⁾⁾/Γₙ⁽⁽⁾⁾⁾ ≈ 0.75. Note also that, as λ₀/d decreases, higher-order NFGMs appear and couple to the ring. For instance, HE₃₁ the cutoff wavelength of which is around 1.3d, couples to REMs n = ±2. This results in the complex peak structure observed in Fig. 2c.

The presence of the ONF also reshapes the radiation patterns. In Fig. 4 we plot the intensity radiated by the
<figure><img src="image_3.png" /><figcaption>FIG. 2: Nanoring of $N = 5$ identical two-level atoms with orthoradial dipoles. Total decay rates of the REM $n = 0, 1, 2$ (a) in free space, $\Gamma_n^{(0)}$, and (b) in the presence of the fiber, $\Gamma_n^{(t)}$, expressed in units of single-atom spontaneous emission rate in free-space $\gamma_0$. (c) Ratio of the decay rate of the REM $n = 0, 1, 2$ towards NFGMs, $\Gamma_n^{(g)}$, over the total decay rate of the same REM, $\Gamma_n^{(t)}$. Ratio of the decay rate of the REM $n = 0, 1, 2$ towards NFGMs $\lambda$, $\Gamma_n^{(\lambda)}$, over the total emission rate of the same REM, $\Gamma_n^{(t)}$, for $\lambda = \text{HE}_{11}$ (d), $\text{TE}_{01}$ (e), $\text{TM}_{01}$ (f), $\text{HE}_{21}$ (g). In our calculations, we set $\rho = 1.1 \times a$ whence $d \approx 1.29 \times a$, we restrict ourselves to the range $\frac{\pi}{5}a \le \lambda_0 \le 2\pi a$, and we take into account 15 radiation modes and all the guided modes supported by the fiber in the considered frequency range. To provide a visual reference, we have plotted the unit value of the different ratios in a dashed grey line in each subfigure.</figcaption></figure>

ring in REM $n = 1$ in free space (a,b,c,f) and close to the ONF (c,d,g,h,i) in $xz$-plane at $y = 1.5\rho$ (a,c) and $y = 5\rho$ (b,d), and in $xy$-plane at $z = 1.5\rho$ (e,g), $z = 5\rho$ (f,h) and $z = 20\rho$ (i). We choose the value $\frac{\lambda_0}{d} \approx 3.4$ for which the spontaneous photon is mainly emitted towards the fundamental NFGM, $\text{HE}_{11}$ (see Fig. 2d). Intensity patterns Figs. 4(a,b) and Figs. 4(c,d) observed in free-space and in the vicinity of the nanofiber, respectively, have the same global structure. The presence of the nanofiber, however, induces a spatial modulation in Figs. 4(c,d) which results from the interference between (i) the light emitted by the ring and directly propagating through free-space towards the observation plane and (ii) the light reaching the observation plane after reflection onto the nanofiber. At the chosen distances $y = 1.5\rho$ and $y = 5\rho$ the guided modes are too weak to play a significant role. By contrast, the intensity pattern observed in the $xy$-plane tends to focus around the fiber (Figs. 4 g-i) compared to the free-space case (Figs. 4 e-f). In particular,

far from the ring, this pattern is strongly dominated by the fundamental NFGM $\text{HE}_{11}$ as clearly seen in Fig. 4i.

As already mentioned at the beginning of this section, nanorings of higher $N$'s (up to $N = 10$) present the same qualitative behavior as the ring considered above in the collective regime. The main difference is the appearance of new REMs $n \ge 3$ which are strongly subradiant in the collective regime of interest, i.e. $\frac{\lambda_0}{d} \ge 2$, as previously observed in free-space [52], and therefore do not play any role in the emission of the ring. In quantum network scenarios, such modes could, however, serve as stable memory states onto which guided photonic excitations could be mapped after being collected by REMs strongly coupled to NFGMs.

Finally, in the Appendix, we present the results we obtained for REM $n = 0$ as well as for longitudinal atomic dipoles (Fig. 1b).
<figure><img src="image_4.png" /><figcaption>FIG. 3: Dispersion relation $\frac{\beta}{k_0} = f(\frac{\lambda_0}{d})$ of the guided modes of a silica ONF ($n_{fiber} = 1.45$). We only label the first few modes relevant to our study. We set $\frac{\ell}{a} = 1.1$ and restrict ourselves to the range $\frac{\pi a}{5} \le \lambda_0 \le 2\pi a$.</figcaption></figure>

### III. TWO RINGS

We now consider the case of two identical rings $(I, II)$ positioned at the lateral distance $\Delta z$ from each other (Fig. 1c). The atoms of the lower and upper rings are labeled from 1 to $N$ and from $(N+1)$ to $2N$, respectively. Assuming the two-ring system is initially singly-excited, it evolves according to the same effective Hamiltonian, $H_{eff}$, as in the previous case, with the difference that the sums over $i$ and $j$ in Eq.(1) now run from 1 to $2N$. Defining the REMs of the first and second rings as $|\Psi_n^{(I)}\rangle \equiv \frac{1}{\sqrt{N}} \sum_{j=1}^N e^{i\frac{2\pi n(j-1)}{N}} |e_j\rangle$ and $|\Psi_n^{(II)}\rangle \equiv \frac{1}{\sqrt{N}} \sum_{j=1}^N e^{i\frac{2\pi n(j-1)}{N}} |e_{j+N}\rangle$, one shows that

$$H_{\text{eff}} |\Psi_n^{(I,II)}\rangle = -\frac{3\pi\hbar\gamma_0}{k_0} \left[ \lambda_n |\Psi_n^{(I,II)}\rangle + \nu_n |\Psi_n^{(II,I)}\rangle \right]$$

with $\lambda_n$'s given in Eq. (2) and

$$\nu_n = \frac{1}{N} \sum_{k,l=1}^{N} g_{k+N,l} e^{-i \frac{2\pi n(k-l)}{N}} \quad (4)$$

In the basis $\{|\Psi_n^{(I)}\rangle_{n=[-\frac{N-1}{2}]...[\frac{N-1}{2}]}\rangle \cup \{|\Psi_n^{(II)}\rangle_{n=[-\frac{N-1}{2}]...[\frac{N-1}{2}]}\rangle$, $H_{eff}$ hence splits into four diagonal blocks (see Figs. 5 a-c): the two blocks on the diagonal are identical and contain the $\lambda_n$'s, the off-diagonal blocks are identical and contain the $\nu_n$'s. An excitation initially stored in a given REM $n$ of one ring is therefore mapped into the same REM $n$ of the other ring, and back and forth until it is lost. Contrary to the $\lambda_n$'s, the coupling terms $\nu_n$'s depend on the ring-ring

separation, $\Delta z$. At short distance (Fig. 5a), $\nu_n$'s include free-space, radiation, and guided mode contributions. As $\Delta z$ increases (Fig. 5b), the free-space and radiation mode components progressively vanish and the $\|\nu_n\|$'s decrease. Finally, at large separation $\Delta z$, only NFGMs contribute significantly. In the case shown in Fig. 5 – orthoradial atomic dipoles and $\lambda_0 = 3.4d$ – since only HE$_{11}$ exists which couples to REMs $n = \pm 1$, only the terms $\nu_n=\pm 1$ remain (Fig. 5c). Similar observations can be made in other cases, e.g. at $\lambda_0 = 1.8d$ for which the ONF supports more guided modes, or with dipoles oriented along the ONF axis (see Appendix).

The two-ring-system REMs are formed as the symmetric and antisymmetric superpositions of single-ring REMs of same $n$, i.e. $|\Psi_n^{(\pm)}\rangle \equiv \frac{1}{\sqrt{2}} (|\Psi_n^{(I)}\rangle \pm |\Psi_n^{(II)}\rangle)$, for $n = [-\frac{N-1}{2}]...[\frac{N-1}{2}]$. The associated eigenvalues are $-\frac{3\pi\hbar\gamma_0}{k_0}\mu_n^{(\pm)}$ with

$$\mu_n^{(\pm)} \equiv (\lambda_n \pm \nu_n) = \frac{1}{N} \sum_{k,l=1}^{N} e^{-in(\theta_k - \theta_l)} (g_{kl} \pm g_{k+N,l})$$

and the decay rates are $\Gamma_n^{(\pm)} \equiv -\frac{6\pi\hbar\gamma_0}{k_0} \text{Im}\mu_n^{(\pm)}$. In Fig. 5, we plot the decay rates of the REMs $|\Psi_n^{(\pm)}\rangle$ in free space, $\Gamma_n^{(0,\pm)}$ (d), and in the presence of the nanofiber, $\Gamma_n^{(\pm)}$ (e), over the free-space single-atom spontaneous emission rate, $\gamma_0$, as functions of $\frac{\Delta z}{a}$ for $\lambda_0 = 3.4d$ (inserts show the variations over a wider range of $\Delta z$). In the (unphysical) limit $\Delta z \to 0$, the two rings merge into a single one and $\nu_n \to \lambda_n$ for all $n$'s, whence $\Gamma_n^{(-)} \to 0$ and $\Gamma_n^{(+)} \to 2\Gamma_n$. As $\Delta z$ increases, decay rates oscillate around their single-ring value, $\Gamma_n^{(0)}$ in free space (d) and $\Gamma_n^{(t)}$ in the vicinity of the fiber (e). These oscillations vanish at large separation in free-space (insert of d) as well as in the vicinity of the fiber (insert of e) except for the REMs ($n = 1, \pm$). Indeed, as mentioned above, the oscillating parts in $\Gamma_n^{(\pm)}$'s are the coupling terms $\nu_n$'s. The latter vanish when $\Delta z \gg a$ for those REMs which couple only to free-space (Fig. 5d) and radiation modes (REMs $n = 0, 2$ in Fig. 5e). By contrast, they remain nonzero for REMs which couple to NFGMs, i.e. here REMs $n = 1$ coupling to HE$_{11}$ (Fig. 5e). Moreover, for free-space and radiation mode contributions, the spatial period of these oscillations is the wavelength $\lambda_0$ while for NFGM contributions it coincides with $\frac{2\pi}{\beta}$, $\beta$ being the propagation constant which satisfies the dispersion relation (Fig. 3). In the case considered in Fig. 5e, REMs ($n = 0, 2; \pm$) couple only to radiation and free-space modes and their decays exhibit damped oscillations with the period $\lambda_0$, while REMs ($n = 1; \pm$) couple predominantly to NFGMs HE$_{11}$ and show persistent oscillations with the period $\frac{2\pi}{\beta_{HE11}} \approx 0.95\lambda_0$. Note that, though not visible in Fig. 5, beatings can be observed between free-space/radiation and guided mode contributions at moderate separation $\Delta z$ (see Appendix).
<figure><img src="image_5.png" /></figure>

FIG. 4: Nanoring of $N = 5$ identical two-level atoms with orthoradial dipoles. Radiation patterns of the nanoring prepared in the REM $n = 1$ in free space (a,b,e,f) and in the vicinity of an ONF (c,d,g,h,i). The 2D patterns are represented in the $xz$-plane at $y = 1.5\rho$ (a,c) and $y = 5\rho$ (b,d), and in the $xy$-plane at $z = 1.5\rho$ (e,g), $z = 5\rho$ (f,h) and $z = 20\rho$ (i). Intensity is expressed in arbitrary units. The marks of planes $z = 1.5\rho$ and $z = 5\rho$ appear in dotted white lines in a-d, and the marks of the plane $y = 1.5\rho$ appear in dotted white lines in e-i. In our calculations, we set $\rho = 1.1 \times a$ whence $d \approx 1.29 \times a$, we choose $\frac{\lambda_0}{d} \approx 3.4$ which corresponds to the maximal ratio $\frac{\Gamma_{n=1}^{(HE_{11})}}{\Gamma_{n=1}^{(t)}}$ (Fig. 2d) and we take into account 15 radiation modes and all the guided modes supported by the fiber at the considered frequency.

Finally, we observe in Fig. 5e that at $\frac{\Delta z}{a} = \frac{2\pi m}{\beta a} \approx 4.2m$ for $m \in \mathbb{Z}$, REMs ($n=1; \pm$) are slightly super ($\Gamma_{\text{sup}} \approx 1.1\gamma_0$) and subradiant ($\Gamma_{\text{sub}} \approx 0.2\gamma_0$), respectively, and the opposite at $\frac{\Delta z}{a} = \frac{2\pi}{\beta a} (m + \frac{1}{2})$. In addition, we note that one can easily switch from $|\Psi_{n=1}^{(\pm)}\rangle$ to $|\Psi_{n=1}^{(\mp)}\rangle$ by imposing a $\pi$ phase shift on the excited state of one ring. Therefore, by adjusting the ring separation $\Delta z$, one can map an excitation from the NFGM $\text{HE}_{11}$ into the superradiant branch of REM $n=1$ before transferring it into the subradiant state via the application of a laser beam on one of the rings, hence protecting it from losses towards radiation modes for about $\tau \approx 5\gamma_0^{-1}$. The excitation lifetime enhancement we obtain here is indeed moderate. Our findings, however, suggest that larger – though finite – arrays of nanorings should give access to a high level of protection while offering switchable, selective, and reliable communication channels via guided modes.

## IV. CONCLUSION

The collective emission of one and two nanorings comprising of $N = 5$ two-level atoms surrounding a silica ONF was theoretically investigated. The radiation eigenmodes of a single ring were shown to couple only to specific guided modes according to their symmetry, while the emission towards radiation modes can be greatly suppressed by properly tuning the parameters of the sys-

tem. Two rings can couple even at a large separation distance via guided modes, and this results in an enhanced sub/superradiant character of the radiation eigenmodes with respect to the single-ring case. We believe that the combination of the mode selectivity offered by the single-ring configuration with the guided mode-assisted long-range collective coupling of arrays of such rings is the optimal combination. holds great potential for efficient and versatile quantum photonic memories and networks. While an experimental realization of the proposed setup is challenging, it is worth investigating using either a system of optical tweezers with specific polarizations to create five traps around the ONF or, alternatively, exploring higher order mode interference that could lead to such a configuration by emptying all other traps along the length of the nanofiber via an optical pushing beam.

Other configurations including more atomic levels and more rings with tilted angular distributions will be considered in the future for they may give rise to interesting effects, including enhanced chiral coupling of guided light with the atomic system. The presumably highly mode-selective reflection and transmission properties of a stack of nanorings will also be addressed. Finally the multi-excitation regime and the potential use of the nanoring configuration for the implementation of two-photon gates shall be investigated.
<figure><img src="image_6.png" /><figcaption>FIG. 5: Coupling of two identical nanorings of $N = 5$ two-level atoms with orthoradial dipoles. Lower left $N \times N$ block in the effective $2N \times 2N$ Hamiltonian expressed in the single-ring eigenmode basis for $\lambda_0 = 3.4$ d and different values of the lateral distance between the rings : (a) $\Delta z = a$, (b) $\Delta z = 10a$, (c) $\Delta z = 20a$. Inserts show the full effective Hamiltonian, $H_{\text{eff}}$. Terms of $H_{\text{eff}}$ are plotted in arbitrary units. Decay rates of the two-ring REMs $(\pm, n = 0, 1, 2)$ in free-space, $\Gamma_n^{(0,\pm)}$ (d), and in the presence of the ONF, $\Gamma_n^{(\pm)}$ (e), over the single-atom spontaneous emission rate in free space, $\gamma_0$, are plotted as functions of $\frac{\Delta z}{a}$ for $\lambda_0 = 3.4$ d. Inserts show the same functions plotted on a larger range of $\frac{\Delta z}{a}$.</figcaption></figure>

## Appendix A: Supplementary numerical results

In this Appendix we briefly present complementary numerical results we obtained in cases not considered in the main text, i.e. orthoradial dipoles at $\lambda_0 \approx 1.8d$ (Sec. A 1) and dipoles along the ONF axis (Sec. A 2).

### 1. Orthoradial dipoles, $\lambda_0 = 1.8d$

We first consider a single ring of 5 atoms with orthoradial dipoles and transition wavelength $\lambda_0 = 1.8d$. In that case, the highest relative coupling to NFGMs is observed for the REM $n = 0$ which emits a photon into the mode $TE_{01}$ with a probability $\approx 0.9$ (see Fig. 2e). In Fig. 6 we plot the corresponding radiation patterns in

free-space (a,b,e,f) and close to the ONF (c,d,g,h), in $xz$-plane at $y = 1.5\rho$ (a,c) and $y = 5\rho$ (b,d), and in $xy$-plane at $z = 1.5\rho$ (e,g) and $z = 5\rho$ (f,h). As in the case examined in the main text, the presence of the ONF induces interference fringes (Figs. 6c,d) not observable in free-space (Figs. 6a,b). The intensity pattern observed in the $xy$-plane tends to concentrate around the fiber (Figs. 6g,h) compared to the free-space case (Figs. 6e-f). Far from the ring, this pattern is strongly dominated by the donut-like NFGM $TE_{01}$ (Fig. 6h).

We now address the two-ring configuration. In Figs. 7a,b we plot the lower left $N \times N$ block in the effective $2N \times 2N$ Hamiltonian which contains the coupling terms $\nu_n$'s on the diagonal. At large separation $\Delta z = 10a$ (Fig. 7b), $\nu_n$'s contain only NFGMs contributions, here mainly due to $TE_{01}$ which couples single-ring REMs $n = 0$.
<figure><img src="image_7.png" /><figcaption>FIG. 6: Nanoring of $N = 5$ identical two-level atoms with orthoradial dipoles. Radiation patterns of the nanoring prepared in the REM $n = 0$ in free space (a,b,e,f) and in the vicinity of an ONF (c,d,g,h). The 2D patterns are represented in the $xz$-plane at $y = 1.5\rho$ (a,c) and $y = 5\rho$ (b,d), and in the $xy$-plane at $z = 1.5\rho$ (e,g) and $z = 5\rho$ (f,h). Intensity is expressed in arbitrary units. In our calculations, we set $\rho = 1.1 \times a$ whence $d \approx 1.29 \times a$, we choose $\frac{\lambda_0}{d} \approx 1.8$ which corresponds to the maximal ratio $\frac{\Gamma_{n=0}^{(TE01)}}{\Gamma_{n=0}^{(TE02)}}$ (Fig. 2e) and we take into account 15 radiation modes and all the NFGMs existing at the considered frequency.</figcaption></figure>

to each other, therefore the term $\nu_0$ dominates. But, contrary to the case examined in the main text, other NFGMs also exist, i.e. $HE_{11}$ and $HE_{21}$, which couple single-ring REMs $n = \pm 1$ and $n = \pm 2$, respectively, and $\nu_{\pm 1, \pm 2}$ therefore do no completely vanish here. As a consequence, the spatial modulation of the two-ring system decay rates, $\Gamma_n^{(\pm)}$, is observed for all REMs, $|\Psi_n^{(\pm)}\rangle$, in the presence of the ONF even at large separation $\Delta z$ (Fig. 7d) while it tends to vanish in free-space (Fig. 7c). It is also worth noticing that the period of these modulations is the same for all REMs in free-space, i.e. $\lambda_0 \approx 1.8d \approx 2.3a$ (Fig. 7c), while it varies from one REM to another in the presence of the ONF (Fig. 7d). Indeed, as underlined in the main text, at large separation $\Delta z$, the couplings $\nu_n$'s are dominated by the NFGM contributions and therefore oscillate at the corresponding period $\frac{2\pi}{\beta}$, $\beta$ being the propagation constant of the relevant NFGM satisfying the dispersion relation (Fig. 3). For $\lambda_0 = 1.8d$ we have $\beta \approx (1.05, 1.3, 1) \times k_0$ for NFGMs $TE_{01}$, $HE_{11}$ and $HE_{21}$, respectively, which couple to REMs $n = 0, 1, 2$, respectively. The respective periods of oscillations of the decay rate $\Gamma_{n=0,1,2}^{(\pm)}$ are therefore expected to be $\frac{2\pi}{\beta} \approx (\frac{1}{1.05}, \frac{1}{1.2}, 1) \times \lambda_0 \approx (2.2, 1.8, 2.3) \times a$, in good agreement with Fig. 7d. Moreover, for REMs ($n = 1, \pm$) at short separation $\Delta z \le 10a$ beatings appear between the free-space mode component – of period $\lambda_0 \approx 2.3a$ – and the NFGM $HE_{11}$ contribution – of period $\frac{2\pi}{\beta} \approx 1.8a$. These beatings are observable because (i) the period of oscillations differ significantly, (ii) the free-space and guided mode components remain comparable

on a sufficiently wide range before unguided modes vanish. By contrast, such beatings cannot be detected with REMs $n = 0, \pm 2$.

## 2. Dipoles along the fiber axis

We consider the case of atomic dipoles oriented along the fiber axis chosen as the $z$-axis. As in the orthoradial orientation case, the spontaneous emission rates of single-ring REMs $n = 0, 1, 2$ exhibit the same peak structure in free-space (Fig. 8a) and in the presence of the fiber (Fig. 8b), though the strongly collective behavior range is shifted towards higher values of the ratio $\frac{\lambda_0}{d}$. Moreover, the emission features of the atomic ring are enhanced, i.e. peaks are higher and minima are deeper. The probability for the atomic ring to emit towards NFGMs, whichever REM it is initially prepared in, is globally weaker than in the orthoradial dipole configuration (Figs 8c-g). Moreover, REMs $n = 0, 1, 2$ specifically emit towards NFGMs $TM_{01}$, $HE_{11}$ and $HE_{21}$, respectively, as shown in Fig. 8d-g. While emission towards the fundamental NFGM $HE_{11}$ is possible at any frequency, NFGMs $TM_{01}$ and $HE_{21}$, however, have cut-off wavelengths, $\frac{\lambda_0}{d} \approx 2.1$ and $\approx 1.4$, respectively, above which they are not supported by the fiber. The probabilities for the ring prepared in REMs $n = 0, 1, 2$ to emit towards the NFGMs $TM_{01}$, $HE_{11}$ and $HE_{21}$, respectively, reach their maximal values $\approx 0.20, 0.44, 0.10$ around $\frac{\lambda_0}{d} \approx 1.40, 1.90, 1.25$, respectively, at which they are subradiant, i.e. $\frac{\Gamma_{n=0,1,2}^{(t)}}{\gamma_0} \approx 0.40, 0.30, 0.25$. By con-
<figure><img src="image_8.png" /><figcaption>FIG. 7: Coupling of two identical nanorings of $N = 5$ two-level atoms with orthoradial dipoles. Lower left $N \times N$ block in the effective $2N \times 2N$ Hamiltonian expressed in the single-ring eigenmode basis for $\lambda_0 = 1.8$ d and different values of the lateral distance between the rings : (a) $\Delta z = a$, (b) $\Delta z = 10a$. Inserts show the full effective Hamiltonian, $H_{\text{eff}}$. Terms of $H_{\text{eff}}$ are plotted in arbitrary units. Decay rates of the two-ring REMs ($\pm, n = 0, 1, 2$) in free-space, $\Gamma_n^{(0,\pm)}$ (c), and in the presence of the ONF, $\Gamma_n^{(\pm)}$ (d), over the single-atom spontaneous emission rate in free space, $\gamma_0$, are plotted as functions of $\frac{\Delta z}{a}$ for $\lambda_0 = 1.8$ d. Inserts show the same functions plotted on a larger range of $\frac{\Delta z}{a}$.</figcaption></figure>

trast, NFGM TE$_{01}$ does not couple to the atomic ring (Fig. 8b). The selective coupling of REMs to specific NFGMs as well as the existence of peaks in Figs. 8c-g are explained by the same kind of arguments as provided in Sec. II. In particular, since atomic dipoles now point along the z-axis, they fully couple to NFGM TM$_{01}$, while they are orthogonal to NFGM TE$_{01}$.

The radiation patterns of the single-ring REMs $n = 1, 0$ are plotted in Figs. 9 and 10 at wavelengths $\frac{\lambda_0}{d} \approx 1.8, 1.3$ at which they strongly couple to NFGMs HE$_{11}$ and TM$_{01}$, respectively. These plots present the same

features as in the orthoradial dipole configuration. Afar from the ONF, the patterns observed in $xz$-plane have the same shape as in free-space though modulated by the interference between the directly transmitted and reflected components of the light emitted by the ring. The patterns observed in $x, y$-plane show the persistence of light concentrated around the fiber axis even at long distance from the ring, by contrast to the free-space case, manifesting the contribution of NFGMs HE$_{11}$ and TM$_{01}$ in the spontaneous emission of REMs $n = 1, 0$, respectively.
<figure><img src="image_9.png" /><figcaption>FIG. 8: Nanoring of $N = 5$ identical two-level atoms with dipoles oriented along the fiber axis. Total decay rates of the REM $n = 0, 1, 2$ (a) in free space, $\Gamma_n^{(0)}$, and (b) in the presence of the fiber, $\Gamma_n^{(t)}$, expressed in units of single-atom spontaneous emission rate in free-space $\gamma_0$. (c) Ratio of the decay rate of the REM $n = 0, 1, 2$ towards NFGMs, $\Gamma_n^{(g)}$, over the total decay rate of the same REM, $\Gamma_n^{(t)}$. Ratio of the decay rate of the REM $n = 0, 1, 2$ towards NFGMs $\lambda, \Gamma_n^{(\lambda)}$, over the total emission rate of the same REM, $\Gamma_n^{(t)}$, for $\lambda = \text{HE}_{11}$ (d), $\text{TE}_{01}$ (e), $\text{TM}_{01}$ (f), $\text{HE}_{21}$ (g). In our calculations, we set $\rho = 1.1 \times a$ whence $d \approx 1.29 \times a$, we restrict ourselves to the range $\frac{\pi}{5}a \le \lambda_0 \le 2\pi a$, and we take into account 15 radiation modes and all the guided modes supported by the fiber in the considered frequency range. To provide a visual reference, we have plotted the unit value of the different ratios in a dashed grey line in each subfigure.</figcaption></figure>

We address the two-ring configuration in Figs. 11, 12. At $\lambda_0 = 1.8d$ (resp. $\lambda_0 = 1.3d$), $\text{HE}_{11}$ and $\text{HE}_{21}$ (resp. $\text{TM}_{01}$ and $\text{HE}_{21}$) are the NFGMs with highest contributions to ring-ring interaction which couple the REMs $n = \pm 1$ and $n = \pm 2$ (resp. $n = 0$ and $n = \pm 2$) of one ring to the same REMs of the other ring. As a consequence, the terms $\nu_{\pm 1, \pm 2}$ (resp. $\nu_{0, \pm 2}$) dominate at large separation $\Delta z$, see Figs 11a,b (resp. Figs. 12a,b). As in the orthoradial case, decay rates, $\Gamma_n^{(\pm)}$, oscillate with $\Delta z$ in the presence of the ONF, even at large separation (Figs. 11d,12d), while these oscillations disappear for increasing $\Delta z$ in free-space (Figs. 11c,12c). These modulations have the same period, $\lambda_0$, for all REMs in free-space. By contrast, in the presence of the ONF, each REM has its own period, $\frac{2\pi}{\beta}$, imposed by the NFGM contribution. Furthermore, in Figs. 11d and 12d we observe beatings between NFGM contributions and free-space modes at

short distance, typically $\Delta z \le 8a$ – out of that range free-space mode contribution to couplings $\nu_n$'s becomes negligible for all REMs. In Fig. 12d, we observe beatings for REMs ($n = 2, \pm$) outside the range where free-space modes substantially oscillate. These beatings are indeed due to the interference of NFGMs $\text{HE}_{21}$ with radiation modes, particularly the radiation mode with azimuthal number $\nu = \pm 3$, as we could check numerically, the contribution of which remains comparable with the guided mode contribution on a wider range.

## ACKNOWLEDGMENTS

This research was funded in part by l'Agence Nationale de la Recherche (ANR), Project ANR-22-CE47-0011. For the purpose of open access, the authors have applied a CC-BY public copyright licence to any Author Accepted
<figure><img src="image_10.png" /><figcaption>FIG. 9: Nanoring of $N = 5$ identical two-level atoms with dipoles oriented along the fiber axis. Radiation patterns of the nanoring prepared in the REM $n = 1$ in free space (a,b,e,f) and in the vicinity of an ONF (c,d,g,h). The 2D patterns are represented in the $xz$-plane at $y = 1.5\rho$ (a,c) and $y = 5\rho$ (b,d), and in the $xy$-plane at $z = 1.5\rho$ (e,g) and $z = 5\rho$ (f,h). Intensity is expressed in arbitrary units. In our calculations, we set $\rho = 1.1 \times a$ whence</figcaption></figure>

$d \approx 1.29 \times a$, we choose $\frac{\lambda_0}{d} \approx 1.8$ which corresponds to the maximal ratio $\frac{\Gamma_{n=1}^{(HE11)}}{\Gamma_{n=1}^{(f)}}$ (Fig. 8d) and we take into account 15 radiation modes and all the NFGMs existing at the considered frequency.

<figure><img src="image_11.png" /><figcaption>FIG. 10: Nanoring of $N = 5$ identical two-level atoms with dipoles oriented along the fiber axis. Radiation patterns of the nanoring prepared in the REM $n = 0$ in free space (a,b,e,f) and in the vicinity of an ONF (c,d,g,h). The 2D patterns are represented in the $xz$-plane at $y = 1.5\rho$ (a,c) and $y = 5\rho$ (b,d), and in the $xy$-plane at $z = 1.5\rho$ (e,g) and $z = 5\rho$ (f,h). Intensity is expressed in arbitrary units. In our calculations, we set $\rho = 1.1 \times a$ whence</figcaption></figure>

$d \approx 1.29 \times a$, we choose $\frac{\lambda_0}{d} \approx 1.3$ which corresponds to the maximal ratio $\frac{\Gamma_{n=0}^{(TM01)}}{\Gamma_{n=0}^{(f)}}$ (Fig. 8f) and we take into account 15 radiation modes and all the NFGMs existing at the considered frequency.
<figure><img src="image_12.png" /><figcaption>FIG. 11: Coupling of two identical nanorings of $N = 5$ two-level atoms with dipoles oriented along the fiber axis. Lower left $N \times N$ block in the effective $2N \times 2N$ Hamiltonian expressed in the single-ring eigenmode basis for $\lambda_0 = 1.8$ d and different values of the lateral distance between the rings : (a) $\Delta z = a$, (b) $\Delta z = 10a$. Inserts show the full effective Hamiltonian, $H_{\text{eff}}$. Terms of $H_{\text{eff}}$ are plotted in arbitrary units. Decay rates of the two-ring REMs $(\pm, n = 0, 1, 2)$ in free-space, $\Gamma_n^{(0,\pm)}$ (c), and in the presence of the ONF, $\Gamma_n^{(\pm)}$ (d), over the single-atom spontaneous emission rate in free space, $\gamma_0$, are plotted as functions of $\frac{\Delta z}{a}$ for $\lambda_0 = 1.8$ d. Inserts show the same functions plotted on a larger range of $\frac{\Delta z}{a}$.</figcaption></figure>

[1] W. Li, D. Brown, A. Vylegzhanin, Z. Shahrabifarahani, A. Raj, J. Du and S. Nic Chormaic, J. Phys. Photonics **6**, 021002 (2024).

[2] M. C. Frawley, A. Petcu-Colan, V. G. Truong and S. Nic Chormaic, Opt. Commun. **285**, 4648-54 (2012).

[3] J. E. Hoffman, F. K. Fatemi, G. Beadie, S. L. Rolston and L. A. Orozco, Optica **2**, 416 (2015).

[4] F. Le Kien, J. Q. Liang, K. Hakuta and V. I. Balykin, Opt. Commun. **242**, 445 (2004).

[5] F. Le Kien, T. Busch, V. G. Truong and S. Nic Chormaic, Phys. Rev. A **96**, 023835 (2017).

[6] T. Nieddu, V. Gokhroo and S. Nic Chormaic, J. Opt. **18**, 053001 (2016).

[7] Ravi Kumar, Vandna Gokhroo, Kieran Deasy, Aili Maimaiti, Mary C. Frawley, Ciarán Phelan, and Síle Nic Chormaic, New J. Phys **17**, 013026 (2015).

[8] R. Yalla, F. Le Kien, M. Morinaga and K. Hakuta, Phys. Rev. Lett. **109**, 063602 (2012).

[9] L. Liebermeister et al, Appl. Phys. Lett. **104**, 031101 (2014).

[10] R. N. Patel, T. Schröder, N. Wan, L. Li, S. L. Mouradian, E. H. Chen and D. R. Englund, Light Sci. Appl. **5**, e16032
<figure><img src="image_13.png" /><figcaption>FIG. 12: Coupling of two identical nanorings of $N = 5$ two-level atoms dipoles oriented along the fiber axis. Lower left $N \times N$ block in the effective $2N \times 2N$ Hamiltonian expressed in the single-ring eigenmode basis for $\lambda_0 = 1.3$ d and different values of the lateral distance between the rings: (a) $\Delta z = a$, (b) $\Delta z = 10a$. Inserts show the full effective Hamiltonian, $H_{\text{eff}}$. Terms of $H_{\text{eff}}$ are plotted in arbitrary units. Decay rates of the two-ring REMs $(\pm, n = 0, 1, 2)$ in free-space, $\Gamma_n^{(0,\pm)}$ (c), and in the presence of the ONF, $\Gamma_n^{(\pm)}$ (d), over the single-atom spontaneous emission rate in free space, $\gamma_0$, are plotted as functions of $\frac{\Delta z}{a}$ for $\lambda_0 = 1.3$ d. Inserts show the same functions plotted on a larger range of $\frac{\Delta z}{a}$.</figcaption></figure>

(2016).

[11] F. Warken, E. Vetsch, D. Meschede, M. Sokolowski and A. Rauschenbeutel, Opt. Express **15**, 11952 (2007).

[12] F. Le Kien, S. S. S. Hejazi, T. Busch, V. G. Truong, and S. Nic Chormaic, Phys. Rev. A **96**, 043859 (2017).

[13] F. Le Kien, V. I. Balykin and K. Hakuta, Phys. Rev. A **70**, 063403 (2004).

[14] J. Fu, X. Yin and L. Tong, J. Phys. B: At. Mol. Opt. Phys. **40**, 4195 (2007).

[15] E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins and A. Rauschenbeutel, Phys. Rev. Lett. **104**, 203603 (2010).

[16] E. Stourm, Y. Zhang, M. Lepers, R. Guérout, J. Robert, S. Nic Chormaic, J. Mølmer, E. Brion, J. Phys. B: At. Mol. Opt. Phys. **52**, 045503 (2019).

[17] E. Stourm, M. Lepers, J. Robert, S. Nic Chormaic, K. Mølmer, E. Brion, Phys. Rev. A **101**, 052508 (2020).

[18] E. Stourm, M. Lepers, J. Robert, S. Nic Chormaic, K. Mølmer, and E. Brion, New J. Phys. **25**, 023022 (2023).

[19] K. S. Rajasree, T. Ray, K. Karlsson, Kristoffer, J. L. Everett and S. Nic Chormaic, Phys. Rev. Res. **2**, 012038 (2020).

[20] A. Vylegzhanin, D. J. Brown, A. Raj, D. F. Kornovan, J. L. Everett, E. Brion, J. Robert, S. Nic Chormaic, Optica Quantum **1**, 6 (2023).

[21] A. S. Sheremet, M. I. Petrov, I. V. Iorsh, A. V. Poshakin-
skiy and A. N. Poddubny, Rev. Mod. Phys. **95**, 015002
(2023).
[22] A. Asenjo-Garcia, M. Moreno-Cardoner, A. Albrecht, H. J. Kimble, and D. E. Chang, Phys. Rev. X **7**, 031024 (2017).

[23] P. Solano, P. Barberis-Blostein, F. K. Fatemi, L. A. Orozco, and S. L. Rolston, Nat. Commun. **8**, 1857 (2017).

[24] N. V. Corzo, J. Raskop, A. Chandra, A. S. Sheremet, B. Gouraud, and J. Laurat, Nature (London) **566**, 359 (2019).

[25] B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, Phys. Rev. Lett. **114**, 180503 (2015).

[26] C. Sayrin, C. Clausen, B. Albrecht, P. Schneeweiss, and A. Rauschenbeutel, Optica **2**, 353 (2015).

[27] N. V. Corzo, B. Gouraud, A. Chandra, A. Goban, A. S. Sheremet, D. V. Kupriyanov, and J. Laurat, Phys. Rev. Lett. **117**, 133603 (2016).

[28] A. S. Sheremet, A. D. Manukhova, N. V. Larionov, and D. V. Kupriyanov, Phys. Rev. A **86**, 043414 (2012).

[29] D. F. Kornovan, A. S. Sheremet, and M. I. Petrov, Phys. Rev. B **94**, 245416 (2016).

[30] H.H. Jen, M. S. Chang, and Y. C. Chen, Sci. Rep. **8**, 9570 (2018).

[31] D. Plankensteiner, C. Sommer, M. Reitz, H. Ritsch, and C. Genes, Phys. Rev. A **99**, 043843 (2019).

[32] Y.-X. Zhang and K. Mølmer, Phys. Rev. Lett. **122**, 203605 (2019).

[33] Y. A. Fofanov, I. M. Sokolov, R. Kaiser, and W. Guerin, Phys. Rev. A **104**, 023705 (2021).

[34] D. Pavolini, A. Crubellier, P. Pillet, L. Cabaret, and S. Liberman, Phys. Rev. Lett. **54**, 1917 (1985).

[35] R. G. DeVoe and R. G. Brewer, Phys. Rev. Lett. **76**, 2049 (1996).

[36] W. Guerin, M. O. Araújo, and R. Kaiser, Phys. Rev. Lett. **116**, 083601 (2016).

[37] J. Rui, D. Wei, A. Rubio-Abadal, S. Hollerith, J. Zeiher, D. M. Stamper-Kurn, C. Gross, and I. Bloch, Nature (London) **583**, 369 (2020).

[38] G. Ferioli, A. Glicenstein, L. Henriet, I. Ferrier-Barbut, and A. Browaeys, Phys. Rev. X **11**, 021031 (2021).

[39] J. A. Needham, I. Lesanovsky and B. Olmos, New J. Phys. **21**, 073061 (2019).

[40] Y.-X. Zhang and K. Mølmer, Phys. Rev. Lett. **125**, 253601 (2020).

[41] D. F. Kornovan, R. S. Savelev, Y. Kivshar, and M. I. Petrov, ACS Photonics **8**, 3627 (2021).

[42] N. Fayard, I. Ferrier-Barbut, A. Browaeys, and J.-J. Gréeffet, Phys. Rev. A **108**, 023116 (2023).

[43] Y. Ke, A. V. Poshakinskiy, C. Lee, Y. S. Kivshar, and A. N. Poddubny, Phys. Rev. Lett. **123**, 253601 (2019).

[44] R. T. Sutherland and F. Robicheaux, Phys. Rev. A **94**, 013847 (2016).

[45] A. Asenjo-Garcia, J. D. Hood, D. E. Chang, and H. J. Kimble, Phys. Rev. A **95**, 033818 (2017).

[46] D. F. Kornovan, N. V. Corzo, J. Laurat, and A. S. Sheremet, Phys. Rev. A **100**, 063832 (2019).

[47] Y.-X. Zhang, C. Yu, and K. Mølmer, Phys. Rev. Res. **2**, 013173 (2020).

[48] A. N. Poddubny, Phys. Rev. A **101**, 043845 (2020).

[49] G. Facchinetti, S. D. Jenkins, and J. Ruostekoski, Phys. Rev. Lett. **117**, 243601 (2016).

[50] R. J. Bettles, S. A. Gardiner, and C. S. Adams, Phys. Rev. A **92**, 063822 (2015).

[51] K. E. Ballantine and J. Ruostekoski, Phys. Rev. Res. **2**, 023086 (2020).

[52] M. Moreno-Cardoner, D. Plankensteiner, L. Ostermann, D. E. Chang, and H. Ritsch, Phys. Rev. A **100**, 023806 (2019).

[53] J. Cremer, D. Plankensteiner, M. Moreno-Cardoner, L. Ostermann, and H. Ritsch, New J. Phys. **22**, 083052 (2020).

[54] R. Holzinger, D. Plankensteiner, L. Ostermann, and H. Ritsch, Phys. Rev. Lett. **124**, 253603 (2020).

[55] M. Moreno-Cardoner, R. Holzinger, and H. Ritsch, Opt. Express **30**, 10779 (2022).

[56] M. Cech, I. Lesanovsky, and B. Olmos, Dispersionless subradiant photon storage in one-dimensional emitter chains, Phys. Rev. A **108**, L051702 (2023).

[57] R. Holzinger, J. S. Peter, S. Ostermann, H. Ritsch, and S. Yelin, Opt. Quantum **2**, 57 (2024).

[58] N. Ustimenko, D. Kornovan, I. Volkov, A. Sheremet, R. Savelev, and M. Petrov, Phys. Rev. A **110**, L011501 (2024).

[59] Fam Le Kien, S. Dutta Gupta, K. P. Nayak, and K. Hakuta, Phys. Rev. A **72**, 063815 (2005).

[60] Fam Le Kien, A. Rauschenbeutel, Phys. Rev. A **95**, 023838 (2017).

[61] F. Le Kien, D. F. Kornovan, S. Nic Chormaic, and T. Busch, Phys. Rev. A **105**, 042817 (2022).

[62] C. T. Tai, Dyadic Green's Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, Piscataway, NJ, 1994).

[63] Fam Le Kien, Thomas Busch, Viet Giang Truong, and Sîle Nic Chormaic, Phys. Rev. A **96**, 023835 (2017).