# Optical Properties of Concentric Nanorings of Quantum Emitters

Verena Scheil  Institut für Theoretische Physik, Universität Innsbruck,
Technikerstr. 21a, A-6020 Innsbruck, Austria  Raphael Holzinger  Institut für
Theoretische Physik, Universität Innsbruck, Technikerstr. 21a, A-6020
Innsbruck, Austria  Maria Moreno-Cardoner  Departament de Física Quàntica i
Astrofísica and Institut de Ciències del Cosmos, Universitat de Barcelona,
Martí i Franquès 1, E-08028 Barcelona, Spain.  Institut für Theoretische
Physik, Universität Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria
Helmut Ritsch  Institut für Theoretische Physik, Universität Innsbruck,
Technikerstr. 21a, A-6020 Innsbruck, Austria

###### Abstract

A ring of sub-wavelength spaced dipole-coupled quantum emitters features
extraordinary optical properties when compared to a one-dimensional chain or a
random collection of emitters. One finds the emergence of extremely subradiant
collective eigenmodes similar to an optical resonator, which feature strong 3D
sub-wavelength field confinement. Motivated by structures commonly appearing
in natural light harvesting complexes, we extend these studies to stacked
concentric multi-ring geometries. We predict that double rings allow to
engineer significantly darker and better confined collective excitation states
over a broader energy band compared to the single ring case. These potentially
enhance weak field absorption and low loss excitation energy transport. For
the specific geometry of the three rings appearing in the natural LH2 antenna
we show that the coupling between the lower double ring structure and the
higher energy blue shifted single ring is very close to a critical value for
the actual size of the molecule. This creates collective excitations with
significant contributions from all three rings as a vital ingredient for
efficient and fast coherent inter-ring transport. Such a geometry thus should
prove useful for the design of sub-wavelength weak field antennas.

###### pacs:

42.50.Ct, 42.50.Nn

##  I Introduction

The optical properties of a quantum emitter, such as its excitation lifetime
and transition frequency, are strongly modified when it is placed close to a
second emitter, due to vacuum fluctuations that mediate dipole-dipole
interactions between them. As a remarkable example, the decay rate of a
collection of emitters separated by subwavelength distances can be enhanced or
suppressed, leading to the well known phenomena of superradiance or
subradiance, respectively [1, 2, 3, 4]. These phenomena are expected to be
strongly enhanced in ordered subwavelength arrays of emitters, where maximal
interference of the scattered fields can be observed [5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27].

Among the different array geometries, a ring-shaped structure formed by
regularly placed emitters has very special optical properties. It has been
shown before [6, 9, 17], that a linear chain of emitters whose inter-particle
distance is smaller than half of the light wavelength supports collective
modes that can guide light and are extremely subradiant with the excitation
lifetime increasing polynomially with the atom number. The lifetime limitation
arises from photon scattering off the ends of the chain. Remarkably, by
joining the ends of the chain to form a closed ring, the lifetime can be
exponentially increased with atom number [17, 28, 29].

Such extraordinary optical properties can be exploited for applications
including efficient energy transfer, single photon sources or light harvesting
[30, 31]. We have previously shown [28, 29], that tailoring the geometry,
orientation and distance between two such nanorings allows for lossless and
high fidelity transport of subradiant excitations, as if the two rings were
two coupled nano-scale ring resonators. Besides subradiant states confining
and guiding light, these nanorings also feature radiant modes whose
corresponding electromagnetic field is strongly focused at its center. By
placing an extra emitter at its center, these modes can be exploited to create
a nano-scale coherent light source with a spectral line width which is
strongly suppressed compared to the single atom decay rate [32]. In this case
the collective optical modes of the ring play the role of the cavity modes and
the central atom acts as the gain medium when incoherently pumped.
Furthermore, if the central emitter is absorptive, the system can be tailored
to achieve a strong absorption cross section way beyond the single atom case,
while the outer ring behaves as a parabolic mirror when illuminated externally
by a coherent light field [33].

In this work, we analyse in detail how the optical properties of two or more
of these nanorings are modified when they are stacked in a concentric way.
Note that this system is radically different compared to the case previously
studied of two rings coupled side by side [28, 29], as it preserves some
rotational symmetry. The study of this geometry is strongly motivated by the
abundant presence in nature of highly efficient photosynthetic complexes
sharing a similar stacked structure [34, 35]. In particular, the active core
photosynthetic apparatus of certain bacteria is formed by chromophores,
featuring an optical dipole transition, which are arranged symmetrically
forming a complex structure of stacked concentric coupled nanorings. Some of
these units are specialized in transforming the absorbed energy into chemical
energy (LH1), while a larger number of them (LH2 and LH3) do not have a
reaction center but efficiently capture and funnel light towards the LH1
units.

In this system, coherence effects between the chromophores have already shown
to play a crucial role in the energy transfer and light harvesting [36, 37,
38]. A natural question is whether collective decay, i.e., superradiance and
subradiance, plays an essential role in this process, and whether nature
chooses a particular geometry in order to optimize its effects. In this work,
we aim at shedding light on this question, by analyzing the optical properties
and exciton dynamics in realistic structures. Furthermore, similar mechanisms
could be in principle exploited for artificial light harvesting. Proving these
concepts could be already possible using state-of-the-art experimental setups,
such as neutral atoms trapped in optical lattices [39, 40, 41, 42], optical
tweezer arrays [43, 44, 45, 46, 47], microwave coupled superconducting q-bits
[48, 49, 50] or solid-state quantum dots [51, 52].

The paper is organized as follows. We first introduce the theoretical
framework to describe a system of dipole-dipole interacting quantum emitters,
and demonstrate that a structure of coupled symmetric nanorings can be
described in a particularly simple form in terms of Bloch eigenmodes. Next, we
summarize the optical properties of single nanorings, which can exhibit
special radiating properties. We then move to study the case of two coupled
nanorings, displaying two energy bands. Thereafter, we apply a similar
analysis to elucidate the radiating properties of a realistic natural light-
harvesting complex (LH2), which contains a closely double ring structure with
a shifted third ring at higher resonance frequencies. Studying this geometry
we find that the rings geometry and size is critically close to the case where
the energy bands of all rings overlap to for common superradiant exciton
states.

##  II Theoretical Framework: Bloch Eigenmodes

Let us consider first a ring-shaped array (or regular polygon) of $N$
identical two-level quantum emitters with minimum inter-particle distance $d$.
The emitters possess a single narrow optical dipole transition around the
frequency $\omega_{0}$ with dipole orientation
$\hat{\boldsymbol{\wp}}_{i}=\sin\theta\cos\phi\,\hat{e}_{\phi,i}+\sin\theta\sin\phi\,\hat{e}_{r,i}+\cos\theta\,\hat{e}_{z}$
($i=1,\dots,N$), where $\hat{e}_{z}$ and $\hat{e}_{r,i(\phi,i)}$ denote unit
vectors along the vertical and radial (tangential) direction defined with
respect to the emitter $i$, respectively [see Fig. 1(a)]. In this work we will
then consider a configuration where two or more of these rings are stacked
concentrically around the $\hat{z}$-axis [see Fig. 1(b)].

All the emitters are dipole-dipole interacting via the electromagnetic field
vacuum fluctuations. After integrating out the optical degrees of freedom in
the Born-Markov approximation [53], the atomic reduced density matrix is
governed by the master equation $\dot{\rho}=-i[H,\rho]+\mathcal{L}[\rho]$
($\hbar\equiv 1$), with the dipole-dipole Hamiltonian

$H=\sum_{ij;i\neq j}\Omega_{ij}\hat{\sigma}^{ge}_{i}\hat{\sigma}^{eg}_{j},$
(1)

and Lindblad operator

$\mathcal{L}[\rho]=\frac{1}{2}\sum_{i,j}\Gamma_{ij}\left(2\hat{\sigma}^{ge}_{i}\rho\hat{\sigma}^{eg}_{j}-\hat{\sigma}^{eg}_{i}\hat{\sigma}^{ge}_{j}\rho-\rho\hat{\sigma}^{eg}_{i}\hat{\sigma}^{ge}_{j}\right),$
(2)

with $i$ and $j$ running over all dipoles. The coherent $\Omega_{ij}$ and
dissipative $\Gamma_{ij}$ dipole-dipole couplings can be written in terms of
the Green’s tensor ${\bf G}({\bf r},\omega_{0})$ in free space:

$\displaystyle\Omega_{ij}$
$\displaystyle=-\frac{3\pi\Gamma_{0}}{k_{0}}\leavevmode\nobreak\
\textrm{Re}\left\\{\hat{\boldsymbol{\wp}}_{i}^{*}\cdot{\bf G}({\bf r}_{i}-{\bf
r}_{j},\omega_{0})\cdot\hat{\boldsymbol{\wp}}_{j}\right\\},$ (3)
$\displaystyle\Gamma_{ij}$
$\displaystyle=\frac{6\pi\Gamma_{0}}{k_{0}}\leavevmode\nobreak\
\textrm{Im}\left\\{\hat{\boldsymbol{\wp}}_{i}^{*}\cdot{\bf G}({\bf r}_{i}-{\bf
r}_{j},\omega_{0})\cdot\hat{\boldsymbol{\wp}}_{j}\right\\},$ (4)

where ${\bf r}_{i}$ is the position of the $i$-th dipole and ${\bf G}({\bf
r},\omega_{0})$ is given by

$\displaystyle{\bf G}({\bf r},\omega_{0})$
$\displaystyle=\frac{e^{ik_{0}r}}{4\pi
k_{0}^{2}r^{3}}\left[(k_{0}^{2}r^{2}+ik_{0}r-1)\mathcal{I}-(k_{0}^{2}r^{2}+3ik_{0}r-3)\frac{{\bf
r}\otimes{\bf r}^{\rm T}}{r^{2}}\right].$ (5)

Here, $k_{0}=\omega_{0}/c=2\pi/\lambda$ is the wavenumber associated with the
atomic transition, $\lambda$ the transition wavelength, and
$\Gamma_{0}=\left|\boldsymbol{\wp}\right|^{2}k_{0}^{3}/3\pi\epsilon_{0}$ is
the decay rate of a single emitter with dipole moment strength
$|\boldsymbol{\wp}|$.

The scattered electromagnetic field can be also retrieved from a generalized
input-output relation [16, 17] once the atomic coherences are known:

$\displaystyle{\bf E}^{+}({\bf
r})=\frac{|\boldsymbol{\wp}|k_{0}^{2}}{\epsilon_{0}}\sum_{i}{\bf G}({\bf
r}-{\bf
r}_{i},\omega_{0})\cdot\hat{\boldsymbol{\wp}}_{i}\hat{\sigma}^{ge}_{i}.$ (6)

Motivated by realistic conditions in natural light harvesting complexes, this
work focuses on the linear optical properties and response of the system at
low light conditions. Therefore, we will restrict our study to the case where
at most a single excitation is present in the system. In this situation, the
first term in the Lindblad operator Eq.(2) (also known as recycling term) only
modifies the ground state population and it is not relevant for the
observables of interest (e.g. scattered fields or excitation population). The
remaining terms in the equation can be recast as an effective non-Hermitian
Hamiltonian:

$\displaystyle H_{\rm
eff}=\sum_{ij}\left(\Omega_{ij}-i\frac{\Gamma_{ij}}{2}\right)\hat{\sigma}^{eg}_{i}\hat{\sigma}^{ge}_{j}.$
(7)

with $\Omega_{ii}=0$. In this situation, the dynamics of the system can then
be fully understood in terms of the collective modes defined by the
eigenstates of $H_{\rm eff}$. Each of these modes have associated a complex
eigenvalue, whose real and imaginary parts correspond to the frequency shift
and decay rate of the collective mode, respectively. As we will see next, for
a symmetric ring-shaped structure these modes have a particularly simple form
as they correspond to Bloch functions.

Figure 1: (a) Schematics of a single ring with lattice constant $d$. Each
emitter features an optical dipole moment (indicated by the red solid arrow)
with orientation
$\hat{\boldsymbol{\wp}}=\sin\theta\cos\phi\,\hat{e}_{\phi}+\sin\theta\sin\phi\,\hat{e}_{r}+\cos\theta\,\hat{e}_{z}$,
where $\theta$ and $\phi$ are the polar and azimuth angle, respectively. The
vertical, radial and tangential unit vectors are indicated by $\hat{e}_{z}$,
$\hat{e}_{r}$ and $\hat{e}_{\theta}$, respectively. The red dashed arrow
denotes the projection of the dipole onto the ring plane. (b) Double ring
structure: two rings of radius $R_{1}$ and $R_{2}$ and lattice constants
$d_{1}$ and $d_{2}$ are stacked concentrically and separated by the vertical
distance $z$. The two rings are in general rotated by an angle $\delta$. The
dashed-line rectangle encloses the two sites (one from each of the rings)
forming a possible unit cell (see main text).

### Bloch Eigenmodes in Rotationally Symmetric Ring Structures

We will consider here ring structures possessing an $N-$fold rotational
symmetry, similarly to those arising in certain natural light harvesting
complexes [34, 35]. In this case, as we will see, the eigenmodes corresponding
to the single excitation manifold will be of the Bloch form, i.e., delocalized
states with well defined angular momentum $m$.

The $N$-fold rotational symmetry enables defining $N$ different unit cells
(for an example, see Fig. 1), which will be denoted by $j=1,\cdots,N$. Each
cell contains in general $d$ dipoles with given orientations
$\hat{\boldsymbol{\wp}}_{j\alpha}$ with $\alpha=1,\cdots,d$. We can then
rewrite Eq.(7) as

$\displaystyle H_{\rm
eff}=\sum_{i,j=1}^{N}\sum_{\alpha,\beta=1}^{d}G_{ij}^{\alpha\beta}\hat{\sigma}^{eg}_{i\alpha}\hat{\sigma}^{ge}_{j\beta},$
(8)

with $G_{ij}^{\alpha\beta}\equiv\hat{\boldsymbol{\wp}}_{i\alpha}^{*}\cdot{\bf
G}({\bf r}_{i\alpha}-{\bf r}_{j\beta})\cdot\hat{\boldsymbol{\wp}}_{j\beta}$.
We note that a structure consisting of several coupled concentric rings with
the same emitter number, each ring being rotationally symmetric, can also be
described within this model. In this case, the unit cell contains one site of
each of the rings, and it has as many components as rings are.

In the following, we demonstrate that the eigenmodes of the coupled structure
are of the Bloch form. The symmetry of the system imposes that the position
and polarization vectors associated with dipole $i\alpha$ transform under a
rotation $\mathcal{U}$ of angle $2\pi/N$ (around the $\hat{z}-$axis) according
to ${\bf r}_{i\alpha}\rightarrow\mathcal{U}{\bf r}_{i\alpha}={\bf
r}_{i+1\alpha}$ and
$\hat{\boldsymbol{\wp}}_{i\alpha}\rightarrow\mathcal{U}\hat{\boldsymbol{\wp}}_{i\alpha}=\hat{\boldsymbol{\wp}}_{i+1\alpha}$.
By noting that ${\bf G}$ is a tensor containing terms proportional to the
identity and to ${\bf r}_{i\alpha}\otimes{\bf r}^{\rm T}_{j\beta}$, and thus
it transforms under the same rotation as ${\bf G}({\bf r}_{i\alpha}-{\bf
r}_{j\beta})\rightarrow\mathcal{U}{\bf G}({\bf r}_{i\alpha}-{\bf
r}_{j\beta})\mathcal{U}^{\dagger}={\bf G}({\bf r}_{i+1,\alpha}-{\bf
r}_{j+1,\beta})$, we can then conclude that
$G_{ij}^{\alpha\beta}=G_{i+1,j+1}^{\alpha\beta}$. Thus, this coupling matrix
can be relabelled as $G_{i+1,j+1}^{\alpha\beta}\equiv G_{\ell}^{\alpha\beta}$,
with $\ell=j-i$ ($\ell=0,\cdots,N-1$), as it is a periodic function only
depending on the difference between the two indices $i$ and $j$. This property
allows to write the Hamiltonian Eq.(8) in terms of Bloch modes as follows:

$\displaystyle H_{\rm eff}$
$\displaystyle=\sum_{i}^{N}\sum_{\ell=0}^{N-1}\sum_{\alpha,\beta=1}^{d}G_{\ell}^{\alpha\beta}\hat{\sigma}^{eg}_{i\alpha}\hat{\sigma}^{ge}_{i+\ell,\beta}$
$\displaystyle=\sum_{m}\sum_{\alpha,\beta=1}^{d}\tilde{G}_{m}^{\alpha\beta}\hat{\sigma}^{eg}_{m\alpha}\hat{\sigma}^{ge}_{m\beta},$
(9)

where $\tilde{G}_{m}^{\alpha\beta}\equiv\sum_{\ell=0}^{N-1}e^{i2\pi
m\ell/N}G_{\ell}^{\alpha\beta}$, and we have defined the creation and
annihilation operators of a collective Bloch mode with well defined angular
momentum $m$:

$\hat{\sigma}^{eg(ge)}_{m\alpha}=\frac{1}{\sqrt{N}}\sum_{\ell=0}^{N-1}e^{(-)i2\pi
m\ell/N}\hat{\sigma}^{eg(ge)}_{\ell\alpha}.$

Here, the periodicity of the wavefunction under a $2\pi$ rotation imposes $m$
to be an integer value, and thus, $N$ linearly independent eigenstates can be
constructed by choosing $m=0,\pm 1,\pm 2,\cdots,\lceil\pm(N-1)/2\rceil$, where
$\lceil\cdot\rceil$ is the ceiling function.

Eq.(9) is not yet in its full diagonal form (except if the unit cell contains
a single dipole), but it already tells us that the angular momentum is a good
quantum number. For each value of $m$, the eigenmodes consist in general of a
superposition of each excited dipole in the unit cell and it can be easily
found by diagonalizing the $d\times d$ complex $\tilde{G}^{\alpha\beta}_{m}$
matrix, leading to $\hat{H}_{\rm
eff}=\sum_{m,\lambda}\left(\Omega_{m\lambda}-i\Gamma_{m\lambda}/2\right)\hat{\sigma}^{eg}_{m\lambda}\hat{\sigma}^{ge}_{m\lambda}$.
Here, $\Omega_{m\lambda}$ ($\Gamma_{m\lambda}$) is the real (imaginary) part
of the eigenvalue associated with Bloch mode $m$ and $\lambda$, whereas
$\hat{\sigma}^{eg}_{m\lambda}$ is the corresponding creation operator.

##  III Optical Properties of Nanorings

###  III.1 Single Nanoring

Let us first summarize some of the most relevant optical properties for a
single ring with $N$ dipoles, i.e., the case where the unit cell contains just
a single dipole. As previously shown in [28, 29] the optical properties of the
ring strongly depend on the size of the ring compared to the light wavelength
and on the dipole orientations. In the following, we focus on two different
limiting regimes: a dense large ring (quasi linear chain) and a small ring
(Dicke limit).

Dense and large ring case (quasi-linear chain limit).– A large ring with a
large number of emitters locally resembles a linear array, and can support
optical modes which do not propagate into the three-dimensional space but are
rather confined and guided through the array. These modes correspond to spin-
waves (Bloch modes) whose quasi-momentum along the chain is larger than the
light wavenumber $k_{0}$. This leads to an evanescent field along the
transverse directions to the array. In the very large ring case, one can
identify the linear momentum $k_{z}\leftrightarrow 2\pi m/Nd$, and the
condition $k_{z}>k_{0}$ sets the value of the angular momentum of the guided
subradiant modes to be $m\gtrsim m_{0}$, with $m_{0}=Nd/\lambda$ associated
with the light line. Moreover, such states can only exist if $d<\lambda/2$, as
the maximum value of $k_{z}$ (or equivalently $m$) is given by the boundary of
the first Brillouin zone.

Despite these similarities, a striking difference between linear and closed
ring configurations is the scaling of the subradiant decay rates with emitter
number. Indeed, by closing the ends of the open chain in a ring structure
losses can be strongly reduced, leading to an exponential suppression of the
decay rates with atom number, in contrast to the polynomial suppression for
the linear chain.

On the other hand, the modes for which $m\lesssim m_{0}$ are in general
radiant. The angular momentum of the brightest state however, strongly depends
on the polarization direction of the atoms. In Fig. 2(a) we have plotted the
collective decay rates versus $m$, for a ring of $N=100$ emitters and
different polarization orientations
$\hat{\boldsymbol{\wp}}_{i}=\left\\{\hat{e}_{z},\hat{e}_{r,i},\hat{e}_{\phi,i}\right\\}$.
For comparison, we also plot the result for an infinitely long linear chain
with the same lattice constant (solid line). Clearly, in this regime, the
radial and transverse (tangential) polarization decay rates tend to those for
the perpendicularly (longitudinally) polarized linear chain, with maximally
bright modes close to the light line $m=m_{0}$ ($m=0$).

Besides studying the radiative properties, it is also interesting to analyse
the sign of the frequency shifts in the collective modes arising due to
dipole-dipole interactions. Figure 2(c) shows the frequency shifts
corresponding to Fig. 2(a). We find that the symmetric $m=0$ mode has a
positive (negative) shift when the dipoles are aligned transversely
(longitudinally). This is not so surprising when thinking of interacting
classical static dipoles which repel (attract) each other if they are aligned
in parallel (in a head-to-tail configuration). Note also that in this regime
the bright states are always energetically lower than the guided subradiant
modes.

Figure 2: Single Ring Optical Properties. (a)-(b) Collective decay rates
$\Gamma_{m}$ and (c)-(d) frequency shifts $\Omega_{m}$ versus angular momentum
$m$, depending on polarization orientation (blue open, blue solid and orange
are for transverse, radial and tangential polarization, respectively). Left
panels correspond to a large ring with $d/\lambda=1/3$ and $N=100$. For
comparison, solid lines show the result for an infinite linear chain with
transverse (blue) and longitudinal (orange) polarization. Right panels are for
$d/\lambda=0.05$ and $N=20$ (Dicke regime). In this case there are only one
(two) bright modes at $m=0$ ($m=\pm 1$) for transverse (tangential and radial)
polarization. For tangential polarization the bright (dark) modes are
energetically low (high), whereas the opposite behavior is found for radial
and transverse polarization.

Small ring case (Dicke limit).– We now focus on a different regime where the
ring diameter is small compared to the light wavelength, i.e.,
$R\lesssim\lambda/2$ (Dicke limit). This regime will be relevant in the study
of natural light harvesting complexes, given the small inter-particle
distances which are few orders of magnitude smaller than the light wavelength.
In this case, the emitters radiate as if they were a single dipole with
effective moment strength and decay rate

$\displaystyle\hat{\boldsymbol{\wp}}_{m,\rm eff}=N^{-1/2}\sum_{\ell}e^{i2\pi
m\ell/N}\hat{\boldsymbol{\wp}}_{\ell},\qquad{}\qquad{}\Gamma_{m}=|\hat{\boldsymbol{\wp}}_{m,\rm
eff}|^{2}\Gamma_{0}.$ (10)

From this expression, one can then easily see that for transverse polarization
only the mode with $m=0$ has a non-vanishing value of the effective dipole
moment $\hat{\boldsymbol{\wp}}_{m,\rm eff}=\sqrt{N}\hat{e}_{z}$, and thus it
is bright and decaying at rate $\Gamma_{m=0}\sim N\Gamma_{0}$. Instead for
tangential or radial polarization there are two bright modes $m=\pm 1$ with
$\hat{\boldsymbol{\wp}}_{m,\rm eff}=\sqrt{N/2}\left(\hat{e}_{x}\pm
i\hat{e}_{y}\right)$ and $\Gamma_{m=\pm 1}\sim N\Gamma_{0}/2$. The remaining
modes are dark with vanishing effective dipole moment and
$\Gamma_{m}\rightarrow 0$. Figure 2(c) shows the decay rates for a ring in
this regime ($d/\lambda=0.05$, $N=20$) with different polarization
orientations
$\hat{\boldsymbol{\wp}}_{i}=\left\\{\hat{e}_{z},\hat{e}_{r,i},\hat{e}_{\phi,i}\right\\}$.
Moreover, note that in general, a ring with polarization
$\hat{\boldsymbol{\wp}}_{i}=\cos\theta\cos\phi\,\hat{e}_{\phi,i}+\cos\theta\sin\phi\,\hat{e}_{r,i}+\sin\theta\,\hat{e}_{z}$
($i=1,\dots,N$) will have three different bright modes $m=0,\pm 1$ with decay
rates $\Gamma_{m=0}=N\Gamma_{0}\sin^{2}\theta$ and $\Gamma_{\pm
1}=(N\Gamma_{0}/2)\cos^{2}\theta$.

In this limit, the collective frequency shifts also acquire a particularly
simple co-sinusoidal form. Indeed, in this regime the interactions
$\Omega_{ij}$ between first neighbouring sites dominate, and one can
approximate

$\Omega_{m}\approx N^{-1}\sum_{\ell}\left(\Omega_{\ell,\ell+1}e^{i2\pi
m/N}+\Omega_{\ell,\ell-1}e^{-i2\pi m/N}\right)=2\Omega_{d}\cos\left(2\pi
m/N\right),$

where we use again the discrete rotational symmetry of the ring. Here the sign
and strength of first-neighbour coupling $\Omega_{d}$ strongly depends on the
polarization direction. For the same general polarization as before,
$\Omega_{d}=-(3\Gamma_{0}/4k_{0}^{3}d^{3})\left[\cos^{2}\theta(3\cos^{2}\phi-\sin^{2}(\pi/N))-1\right]$
[29]. Therefore, the bright modes will be energetically high (low) for
transverse / radial (tangential) polarization, as it is shown in Fig. 2(d) for
the same parameters as before. Moreover, for polarization angles
$\cos\theta\approx 1/\sqrt{3}\cos\phi$ and large number of emitters, a nearly
degenerated flat band emerges, with frequency shifts that basically vanish
[29].

Finally, it is also possible to evaluate the electromagnetic field generated
by one of these eigenmodes, by using Eq.(6). The result will strongly depend
on the angular momentum $m$, polarization orientation, and size of the ring.
For the ring geometry, we find that strongly subradiant modes radiate with
very low intensity basically along the ring plane, while the field is
evanescent in the transverse direction, as shown in the top row of Fig. 6 and
Fig. 7 for a ring of $N=9$ tangentially polarized emitters and $m=4$. Instead,
the brightest modes (which in this case correspond to $m=\pm 1$) exhibit a
strong field at the center of the ring and propagates also transversally to
the ring plane, as shown in the same figures.

###  III.2 Two Coupled Nanorings

Figure 3: Two coupled identical non-rotated nanorings ($\delta=0$). (a)-(c)
Collective decay rates $\Gamma_{m}$ and (d)-(f) frequency shifts $\Omega_{m}$
versus angular momentum $m$, for two coupled rings of $N=9$ emitters each and
$R/\lambda=0.05$. The blue (orange) dashed lines denote the symmetric (anti-
symmetric) eigenmodes. For comparison, the single ring solution for the same
parameters is shown (grey solid line). The two rings are separated by the
vertical distance $Z=0.5R$, and the emitters have transverse, radial or
tangential polarization (left, middle or right panels, respectively). For
transverse (radial and tangential) polarization the symmetric band is lower
(higher) in energy.

We now analyze the case of two rings of radius $R_{1}$ and $R_{2}$ that are
arranged concentrically and separated by a vertical distance $Z$. In general,
we will also allow in the model a general rotation of angle
$\delta\in[0,2\pi/N)$ of one of the rings around the $\hat{z}$-axis (see Fig.
1). In this case the unit cell consists of only two dipoles ($d=2$).

Coupled identical non-rotated rings ($\delta=0)$.– We first focus on the case
of two identical rings ($R_{1}=R_{2}$) concentrically stacked on top of each
other and with no rotation angle $\delta$. Because the two rings are
identical, and due to $\delta=0$, the matrix $\tilde{G}_{m}^{\alpha\beta}$ is
complex symmetric, and the eigenmodes of Eq.(8) can be chosen as the symmetric
and the anti-symmetric superposition of Bloch states corresponding to each
ring with well defined angular momentum $m$, which will be denoted as
$\mathinner{|{\Psi^{\pm}_{m}}\rangle}=(\mathinner{|{m,1}\rangle}\pm\mathinner{|{m,2}\rangle})/\sqrt{2}$
(with
$\mathinner{|{m\alpha}\rangle}\equiv\hat{\sigma}^{eg}_{m\alpha}\mathinner{|{g}\rangle}$).
The corresponding collective frequency shifts and decay rates are then simply
given by $\Omega_{m}^{\pm}=\Omega_{m}\mp\Omega^{\rm inter}_{m}$ and
$\Gamma_{m}^{\pm}=\Gamma_{m}\pm\Gamma_{m}^{\rm inter}$, where $\Omega_{m}$ and
$\Gamma_{m}$ are the frequency shift and decay rate corresponding to a single
ring, whereas $\Omega_{m}^{\rm inter}={\rm Re}[\tilde{G}_{m}^{12}]$ and
$\Gamma_{m}^{\rm inter}=-2{\rm Im}[\tilde{G}_{m}^{12}]$ are the dispersive and
dissipative inter-ring couplings, respectively.

In Fig. 3 we plot for two rings in the Dicke regime ($R/\lambda=0.05$) and
separated by vertical distance $Z=0.5R$ the decay rates and frequency shifts
of the two emerging bands: symmetric $\mathinner{|{\Psi^{+}_{m}}\rangle}$
(orange line) and anti-symmetric $\mathinner{|{\Psi^{-}_{m}}\rangle}$ (blue
line). For comparison, we overlay the result for two independent rings (grey
line). We find that, regardless of the emitters polarization, the anti-
symmetric solution is always more subradiant than the symmetric one. Moreover
the darkest state is $\Psi^{-}_{\max[m]}$, i.e., the anti-symmetric
superposition of the darkest state of a single ring. Looking at the frequency
shifts, we find that the behavior with angular momentum $m$ is similar to that
for the single ring case, but shifted in energy. In particular, the symmetric
band is shifted to lower energies (higher energies) for transverse (tangential
and radial) polarization of the emitters. This fundamental difference in the
energy shift sign can be intuitively understood in analogy to the energy of
two interacting static dipoles. For the case with transverse polarization, two
closer emitters from the two different rings are in a tail-to-head
configuration, thus decreasing its total energy if they are in phase. Instead,
for the case of tangential and radial polarization the emitters polarization
is parallel, increasing its energy when they have the same phase. In
conclusion, these results show that the polarization of the emitters can
fundamentally modify the optical properties of the emerging bands and
determine the ordering of states in energy, something which is relevant in the
excitation transfer between the different energy bands. In particular the
energy transfer in photosynthetic processes involving dipole interacting
chromophores is understood via H- and J-aggregation. In J-aggregates,
neighboring chromophores are oriented in a head-to-tail arrangement, resulting
in a negative coherent nearest-neighbor coupling $\Omega_{d}$ and the
positioning of the optically allowed ($m=0$) Bloch mode at the bottom of the
energy band, whereas for H-aggregates the orientation is parallel and the
symmetric ($m=0$) mode is positioned at the top of the energy band.

Figure 4: Two coupled identical non-rotated nanorings ($\delta=0$). Scaling of
the most subradiant eigenmode decay rate for two coupled rings (blue) with
$Z/\lambda=0.009$ versus the atom number $N$ of each of the rings. For
comparison, we overlay the most subradiant decay rate for a single ring of $N$
atoms (a) and a single ring of $N_{\rm tot}=2N$ atoms (b) (green) with fixed
inter-particle distance $d/\lambda=1/3$ and transverse polarization. Similar
results are found in case of tangential polarization.

Another interesting property of this system is the scaling of the most
subradiant state decay rate with the atom number $N$. For a fixed inter-
particle distance $d/\lambda$, we show in Fig. 4(a) the decay rate of the most
subradiant state of two coupled rings of $N$ emitters each, compared to that
of a single ring of $N$ emitters (left panel). We observe that, in addition to
a lower decay rate, the double ring structure shows always a stronger
exponential suppression with the atom number compared to a single ring of the
same size and interparticle distance $d$. In Fig. 4(b) we also compare the
double-ring result but with a single ring of $2N$ emitters and the same
density. We find that in this case, for small inter-ring distances $z$ and
ring atom number $N$, the coupling between the two rings is still strong
enough to lead to more subradiance compared to the single ring case with the
same total number of atoms. However, if $N$ is too large, then the single ring
will always support the most subradiant state, as the curvature and therefore
losses will experience a strong suppression as the system approaches an
infinite linear chain, for which it is known that the decay rates are exactly
zero. For this threshold the exponential suppression with $N$ overcomes the
coupling effect between the two rings.

Figure 5: Two coupled identical non-rotated nanorings ($\delta=0$). (a) Most
subradiant decay rate of two coupled rings with $N=9$ emitters and transverse
polarization, as a function of ring constant $d/\lambda$ and inter-ring
distance $z/\lambda$. Subradiant states can exist even beyond the threshold
$d/\lambda<1/2$ and $z/\lambda<1/2$ due to destructive wave interference. (b)
Overlap of the most subradiant eigenmode with the Bloch waves corresponding to
angular momentum $|m|$. The Bloch waves of each ring can form symmetric and
antisymmetric superpositions and it can be seen, that at various distances the
symmetric superposition of $m=0$ Bloch waves can be subradiant. The parameters
are identical to (a) and the overlap oscillates when varying the ring
parameters, as soon as $d,z\gtrsim\lambda/2$.

Interestingly, the most subradiant decay rate does not show a monotonic
behavior with the lattice constant $d/\lambda$ or the inter-ring distance
$z/\lambda$. In 5(a) we plot the most subradiant decay rate versus these two
ratios. We observe that the decay rate oscillates due to wave interference and
that there can still exist subradiance beyond the values $d/\lambda=1/2$ and
$z/\lambda=1/2$. As previously discussed, such subradiant state is always the
anti-symmetric superposition of two Bloch waves of well defined angular
momentum $m$. For small rings such that $d/\lambda<1/2$, the most subradiant
state always corresponds to the superposition of the two most subradiant
states, i.e., $|m|=\lceil(N-1)/2\rceil$. However, for $d/\lambda>1/2$ the
value of $m$ that produces the most subradiant state periodically varies. This
behavior is shown in 5(b), where we have plotted the overlap of the Bloch
waves of particular absolute value of the angular momentum. Additionally the
Bloch waves can be in a symmetric or antisymmetric superposition and even the
symmetric superposition of the symmetric $m=0$ modes can lead to subradiance
at various distances.

We finally discuss the striking differences in the field patterns generated by
the eigenmodes $\mathinner{|{\Psi_{m}^{\pm}}\rangle}$, with $m=0,1,4$. In Fig.
6 and Fig. 7 we plot (middle and bottom rows) the field intensity as a
function of real space position, for two identical coupled concentric rings of
$N=9$ emitters with tangential polarization, lattice constant $d/\lambda=0.1$
and separated by a vertical distance $Z/\lambda=0.2$. For comparison, we have
added in the top row the result for a single ring with the same parameters. We
find that the symmetric superposition shows a pattern which is very similar to
the single ring case. The brightest mode ($m=1$ in this case) shows an
enhanced field intensity along the central axis of the rings. In the symmetric
mode, the field is enhanced in the region between the two rings, whereas in
the anti-symmetric superposition, it shows a striking different pattern with
suppressed field in the region between the two rings.

Coupled unequal rings with rotation ($\delta\neq 0$).– We now consider the
more general case where the two rings can have different radius and are
rotated by an angle $\delta$. Note that in this case the matrix describing the
single excitation manifold $\tilde{G}_{m}^{\alpha\beta}$ is in general not
complex symmetric. However, for the equal radius case ($R_{1}=R_{2}$) in the
Dicke regime, the off-diagonal elements satisfy
$\tilde{G}_{m}^{\alpha\beta}=(\tilde{G}_{m}^{\beta\alpha})^{*}$
($\alpha\neq\beta$). This leads to eigenmodes of the form
$\mathinner{|{\Psi^{\pm}_{m}}\rangle}=(\mathinner{|{m,1}\rangle}\pm
e^{i\eta}\mathinner{|{m,2}\rangle})/\sqrt{2}$ with
$\eta=\textrm{atan}\left[\textrm{Im}\tilde{G}_{m}^{12}/\textrm{Re}\tilde{G}_{m}^{12}\right]$.

The behavior of the eigenmodes and eigenvalues with the rotation angle
$\delta$ is not trivial and strongly depends on the polarization orientation
and inter-particle distances. For transverse polarization and a small vertical
separation between the rings ($Z=0.1R$, $R/\lambda=0.05$ we find a value of
$\delta_{c}\sim 0.15$ for which the frequencies of the two eigenmodes with
$m=\lceil(N-1)/2\rceil$ feature an avoided level crossing. Interestingly, at
this point the nature of the state changes. While for $\delta<\delta_{c}$ the
highest energy state is radiant with $\eta\sim 0$, for $\delta>\delta_{c}$ the
highest energy state becomes subradiant with $\eta\sim\pi$. These features are
shown in Fig. 11 (top panels) and disappear for too small value of $Z$.
Similar results can be found for other values of $m$. Moreover, the decay rate
of the most subradiant state presents a broad minimum around $\pi/N$ and with
$\eta\sim\pi/2$, i.e., when the sites of the second ring lie exactly in
between those of the first ring. At this point and because the interparticle
distances are larger, the frequency shifts are also smaller.

Similar results can be found for other polarization orientations and also when
varying the relative radius between the two rings. As an example, we show in
Fig. 11 (bottom panels) the same analysis for two co-planar rings ($Z=0$) with
tangential polarization and $R_{1}=0.9R_{2}$. As it can be seen in the figure,
in this case there is also an avoided level crossing (inset) at value
$\delta_{c}\sim 0.07$, where the state of the highest energy state changes to
be subradiant. As in the previous case, we find also the broad minimum around
$\delta\sim\pi/N$, where the frequency shifts almost vanish. It is worth
noting that in the natural light harvesting complex LH2 (see next section) the
dipoles of the B850 band are arranged in a similar configuration with rotation
angle $\delta\sim\pi/N$. An intriguing question is whether this is an
accidental coincidence or whether the broad minimum emerging in the decay
rate, which is thus robust against small fluctuations in the emitters
position, can play a relevant role in the energy transfer and the light
harvesting processes.

Figure 6: Two coupled identical non-rotated nanorings ($\delta=0$). Field
intensity pattern versus real space coordinates in units of the transition
wavelength $\lambda_{0}$ (cut at $z=6R$) generated by the eigenmodes with
$m=0,1,4$ as indicated in the panels. Middle and bottom rows correspond to the
symmetric and anti-symmetric eigenmodes, respectively. Top panels are for the
single ring, for comparison. ($N=9$, $d/\lambda=0.1$, $Z/\lambda=0.2$,
tangential polarization.) Figure 7: Two coupled identical non-rotated
nanorings ($\delta=0$). Field intensity pattern versus real space coordinates
(cut at $y=6R$) generated by the eigenmodes with $m=0,1,4$ as indicated in the
panels. Middle and bottom rows correspond to the symmetric and anti-symmetric
eigenmodes, respectively. Top panels are for the single ring, for comparison.
($N=9$, $d/\lambda=0.1$, $Z/\lambda=0.2$, tangential polarization.)  Figure 8:
Two coupled rotated nanorings ($\delta\neq 0$). (Top panels) Two identical
nanorings ($R=0.05\lambda$) with transverse dipole orientation separated by a
vertical distance $Z=0.1R$, depending on the rotation angle
$\delta\in[0,2\pi/18]$: (a) decay rate and (b) frequency shift of the two
eigenmodes with $m=\lceil(N-1)/2\rceil$. An avoided level crossing emerges at
$\delta\sim 0.15$, where the highest energy level changes from being
subradiant to radiant, and from being antisymmetric to symmetric. (Bottom
panels) Two coplanar unequal nanorings ($Z=0$) with radius $R_{1}=0.05\lambda$
and $R_{2}=0.9R_{1}$ and tangential dipole orientation, depending on the
rotation angle $\delta$: (c) decay rate and (d) frequency shift of the two
eigenmodes with $m=\lceil(N-1)/2\rceil$. Similarly as before, an avoided level
crossing (shown amplified in the inset) emerges at $\delta\sim 0.07$, where
the highest energy level changes from being radiant to subradiant, and from
being symmetric to anti-symmetric.

###  III.3 B850 and B800 Bands in LH2

As already anticipated, the study of the optical properties of two (or more)
coupled nanorings is motivated by the existence of similar structures in
nature that enable efficient light harvesting and energy transfer [35, 54, 55,
56, 57, 58, 59, 60, 61, 62, 63, 64]. Indeed, while most biological systems are
soft and disordered, photosynthetic complexes in certain purple bacteria
exhibit crystalline order. The complexes are composed by antenna units that
show a n-fold symmetry [56] which in turn, are arranged forming a maximally
packed hexagonal pattern [65]. Purple bacteria are among the oldest living
organisms and most efficient in turning sunlight into chemical usable energy.
One of the most common species (Rhodopseudomonas Acidophila) contains two well
differentiated types of complexes: a larger one containing the reaction center
where the energy conversion takes place (LH1), and a second one (LH2) which is
more abundant and whose main role is the absorption of photons and efficient
subsequent energy transfer towards the LH1 units. The two complexes are formed
by the same light-absorbing pigments: carotenoids (absorbing wavelengths
ranging from $400$ to $550$ nanometers) and bacteriochlorophyl-a (BChla,
absorbing in the red and infrared). The BChla features a two-level optical
dipole transition around $800-875$ nanometers (depending on the complex).
These pigments are sustained by a hollow cylinder of apoproteins whose
diameter is few tens of Å.

Here we will focus on the LH2 complex and the optical properties displayed by
the BChla. Early x-ray crystallography data [34] together with subsequent
molecular dynamics simulations [66] suggests a ring structure with 9-fold
symmetry. This structure consists of a ring of $9$ emitters maximally
absorbing at $800\textrm{ nm}$ (the so-called $B800$ band) concentrically
arranged and coupled to another two-component ring with 9-fold symmetry (with
a total of $18$ emitters) maximally absorbing at $850\textrm{ nm}$ (the so-
called $B850$ band). The dipoles orientation also preserves the 9-fold
rotational symmetry and are mostly contained in the plane of the ring, except
for a small vertical component (see inset in Fig. 9). Therefore, the whole
structure can be regarded as a ring of $9-$unit cells of $3$ components
(denoted by purple, blue and yellow in the figure).

In the following, we analyse the eigenmodes and collective optical properties
of the two bands ($B800$ and $B850$) using the parameters extracted from [56].
This analysis can be relevant for the understanding of the efficient energy
transfer between the $B800$ and the $B850$ bands, but also for energy transfer
between the LH2 and the LH1 units. Taking into account that the lifetime of
the excited state in the BChla is of the order of nanoseconds, the energy
transfer process is expected to occur at a much faster time scale. Figure 9
shows and compares the decay rates and frequency shifts of the collective
eigenmodes as a function of the angular momentum quantum number $m$,
considering the rings are uncoupled (left) or coupled (right). The dispersive
couplings between the two components of the $B850$ band (denoted by yellow and
blue in the figure) are very large due to the small inter-particle distances,
and of the order of $10^{6}\Gamma$ (being $\Gamma\sim 25\textrm{ MHz}$ the
estimated decay rate of the excited state in the dipole transition). This
leads to the emergence of a two-band structure with large frequency splitting
where the two components of the $B850$ ring strongly hybridize: a higher
energy band which is mostly subradiant, and a lower energy band containing
only two bright modes at $m=\pm 1$. For completeness, we show in Fig. 10 the
excited state population of each of the components for the coupled system
eigenmodes. Clearly, the excitation is delocalized over the two components of
the ring.

In the inset of Fig. 10 we show the small contributions of the lower double
ring configuration to the excited state population of the third band. A
similar behavior emerges in case of the first and second band, where the
$B800$ ring gives a non-vanishing contribution to the population of the first
and second band.

In contrast, the coupling between the $B850$ and $B800$ band (indicated by
purple in the plot) is ten times smaller (of the order $10^{5}\Gamma$),
whereas the energy transition difference is of the order of $10^{7}\Gamma$,
and therefore, the $B800$ band remains mostly decoupled. However, it is worth
noting that after the $B850$ bands are coupled, the higher energy band lies
close to the $B800$ band.

Figure 9: LH2 dipole configuration. (a)-(b) Collective decay rates and (c)-(d)
frequency shifts as a function of angular momentum index $m$ for the LH2
structure (B800 and B850 bands) parameterized according to [56]. Left and
Right panels correspond to uncoupled and coupled rings, respectively. The B850
band consists of a two-component unit cell ring with 9-fold symmetry (denoted
by blue and orange), whereas the B800 band is a single component ring with
9-fold symmetry (denoted by violet). The B800 ring is far in energy and thus
only couples very weakly to the B850 rings. However, the two components of the
B850 band are strongly coupled, due to the reduced inter-particle distance,
what leads to a broad dispersion in the frequency shifts. Two bands emerge: a
darker band which is higher in energy and close to the B800 band, and a
brighter band (with two bright modes corresponding to $m=\pm 1$) which is
lower in energy. This band structure is relevant for the excitation energy
transfer occurring between the B800 and B850 bands. Figure 10: LH2 dipole
configuration. Individual ring occupation propabilities for each of the three
eigenmodes as a function of angular momentum index $m$. Blue and yellow
correspond to the two B850 rings (as indicated in Fig. 9), whereas violet is
the occupation of the B800 ring. Each panel is a different eigenmode,
indicated with the same code color as in Fig. 9.

Finally let us point out a very special property of the naturally occurring
geometry. Indeed it can be seen that the actual geometry is very close to the
critical transition point, where the up shifted eigenstate energies of the
lower double ring just overlap with the upper ring energies. For this in Fig.
11 we plot the corresponding exciton energies as function of the overall size
of the molecule, where we only consider small size variations
$R_{\alpha,i}=\alpha R_{i}$ around the actually measured size. We see that
closely below the value of $\alpha=1$ the energy bands cross and eigenstates
appear which posses similar contributions of all three rings. Close to this
resonance condition any excitation in one of the rings is thus coherently
transported to the other rings in short time. Interestingly the crossing point
depends on the angular index $m$ shifting further away from $\alpha=1$ with
growing $m$. From this sensity behaviour one could expect tune-ability of the
ring properties via the local refractive index or small deformations of the
complex.

Figure 11: LH2 dipole configuration. (a) Frequency shift and (b) ring
occupation probabilities for the third band as function of the overall size of
the molecule at the $m=1,4$ mode. The size of ring $i$ is varied via
$R_{\alpha,i}=\alpha R_{i}$ around the actual size ($\alpha=1$). Solid
(dashed) lines correspond to the $m=1$ ($m=4$) mode. The color code is
equivalent to the one in Fig. 9. Dependent on the mode $m$ the second and
third band as well as the excited state populations cross at
$\alpha_{\mathrm{c}}<1$. For systems with $\alpha<\alpha_{c}$, the third band
is occupied by the B850 ring whereas for $\alpha>\alpha_{c}$ it is occupied by
the B800 ring.

##  IV Conclusions and Outlook

Our calculations show that structures involving multiple concentric rings
exhibit strongly modified exciton properties and in particular, feature
extremely subradiant states with sub-wavelength confined fields. For two
identical rings at close enough distances, we find that the anti-symmetric
superposition of the individual ring radiative modes, which inherits the
angular symmetry of the setup, is always more subradiant than the
corresponding symmetric combination. In particular, the most subradiant states
are obtained by choosing the individual ring darkest eigenmodes. We have shown
that the spontaneous emission of such states decays faster with the emitter
number compared to the single ring case.

Moreover, important radiative properties, such as the ordering in frequency of
the optical modes, can be controlled via a relative rotation or size
differences of two otherwise identical rings. For instance, we find that by
modifying these parameters, the highest energy level changes from being
subradiant to superradiant.

When we apply our model to the specific geometry of the triple ring LH2
structure including the natural distances, energy shifts and dipole
polarization, we find most of the collective modes are extremely dark. Most
interestingly the collective energy shifts from the lower double B850 ring
structure, for which the interparticle distances are very small, is of the
order of the $50$ nm energy shift of the upper ring, so that the energy
spectrum spans almost the full gap between the rings. More specifically, two
bands emerge due to the strong coupling between the two B850 components. A
subradiant band which is higher in energy and close to the B800 band, and a
brighter band which is much lower in energy. The realistic dipole orientations
and distances lead to only two bright modes corresponding to a quasi-symmetric
superposition of the angular momentum $m=1$ and $m=-1$ modes. This emerging
band structure could be helpful for any phonon induced collective energy
transfer processes, which are, of course, beyond our model here, but which we
plan to explore in future work.

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