1. Introduction

Collective radiation effects such as sub- and super-radiance [1–4] in sub-wavelength structures of dipole coupled quantum emitters create growing widespread interest [5–31] as recent experimental advances allow implementing and controlling precise arrays of individual quantum emitters at close distance e.g. in uniformly filled optical lattices [32–35], optical tweezers arrays [36–40], microwave coupled superconducting q-bits [41–43] or solid-state quantum dots [44,45].

Ordered dipole arrays create novel platforms for enhanced atom-light coupling surpassing current limitations of quantum information protocols [20,46,47], precision spectroscopy [48,49] or opto-mechanics [50,51]. Moreover they represent a genuine test bed for fundamental studies of quantum many body states of light and matter [27,49,52,53]. Nature is abundantly engineering complex sub-wavelength scale structures of optical dipoles in common light harvesting complexes [54–60].

Among various designs, arrays forming regular polygons with sub-wavelength inter-particle distance (referred here as nanorings) exhibit exceptional radiative properties [20,61–65]. On the one hand they support extremely subradiant guided modes with a loss exponentially decreasing with atom number [20], allowing for efficient energy transport within a single ring [62] or between two neighboring rings [63,65]. On the other hand they possess collective eigenmodes with a tightly confined field in a sub-wavelength region near their center. Adding gain at the ring center, such nano-rings as optical resonators allow to create coherent laser-like nanoscopic light sources [64].

Here we exhibit how such a ring structure can act like a parabolic mirror antenna [66,67] concentrating incoming radiation in a sub-wavelength volume at its center and strongly enhancing the single photon absorption cross section by placing an impurity there, way beyond the single atom value. Surprisingly one finds that antenna rings of $N=9$ dipoles with an equal center dipole exhibit a distinctively superior performance compared to other antenna atom numbers in several respects. Let us note here, that ring structures in a common form of bacterial light harvesting complexes called LHC2 often appear with 9-fold rotational symmetry, but are comprised of several concentric rings [54–59,68–74]. Interesting previous results show that the dynamics of excitation energy transfer between concentric rings at room temperature are indeed improved by coherence or quantum mechanical coupling between the chromophores forming the ring [70–74].

Analysing the collective eigenmodes reveals a first hint for the astonishing superiority of a nine-atom ring with a central impurity. Only for $N=9$ we find an extremely subradiant mode with the central emitter as the most strongly excited component. Similar sub-radiant states can also be engineered for other ring emitter numbers, when one precisely optimizes their center impurity dipole moment and transition frequency. As a key property, despite being strongly subradiant, these collective states still sufficiently couple to a perpendicular incident plane wave. At resonance, the high-Q center field enhancement of the dark mode creates a large steady state excitation of the central emitter so that the energy can be efficiently absorbed.

Importantly, throughout the absorption process the ring atoms are only weakly excited and will hardly dissipate or re-emit energy. The mechanism resembles a generalized form of cavity anti-resonance spectroscopy with the ring acting as enhancement cavity [75]. Interestingly, the collection efficiency enhancement is even much larger, if the incoming field does not directly excite the central absorber, but only couples to the antenna ring. Any irreversible non-radiative decay from the excited state of the central emitter slowly but efficiently extracts the collected energy without re-emission into free space. Hence the system is a minimalist model for a light-harvesting complex build of a ring shaped antenna and a central absorber as energy dump [54,58–60,76].

2. Antenna model

We model our generic antenna as a regular polygon of $N$ equal point-like two-level emitters with dipole moment $\wp _i = \wp$ at distance $d$ with radius $R$. At its center we add an extra dipole (referred to as “impurity”), with a transition of polarization strength $\wp _I = \sqrt {\Gamma _I/\Gamma _0}~\wp$ detuned from the antenna atoms by $\delta _I$. Energy loss is modelled by an additional incoherent decay channel to an auxiliary state $\left |t\right \rangle$ at rate $\Gamma _T$ (see Fig. 1). All emitters (including the impurity) are interacting via vacuum mediated dipole-dipole interactions which in the Born-Markov approximation leads to the master equation (in the frame rotating at $\omega _0$) [77]:

 

Fig. 1. Scheme of an antenna in form of a regular polygon of $N$ two-level emitters of radius $R$ and distance $d$ coupled to a central impurity which decays from state $\left |e\right\rangle$ either to state $\left |g\right\rangle$ at rate $\Gamma _I$ or to an auxiliary state $\left |t\right\rangle$ via an extra irreversible channel at rate $\Gamma _T$. The impurity $\left |g\right\rangle$-$\left |e\right\rangle$ transition is detuned from the antenna atoms by $\delta _I$. The whole system is uniformly driven.

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3. Sub-radiant modes in the ring-impurity system

A sub-wavelength ring of emitters exhibits extremely subradiant excitation modes whose field vanishes at the center [63] and thus they are decoupled from the center impurity. However, in presence of the impurity one finds a new class of subradiant states with large center occupation and strong coupling to the symmetric bright mode of the outer ring (antenna). When the antenna is excited the center impurity oscillates with opposite phase and almost perfectly cancels the total emitted field creating a so called anti-resonance [75]. The large excitation weight of the central emitter in combination with the suppressed decay here is the key to low loss energy transfer and large absorption cross-section.

For concreteness and simplicity we restrict ourselves to the symmetric case where all emitters are circularly polarized in the ring plane. Nevertheless, equivalent phenomena appear in more general polarization configurations [65], where all emitters equally couple to the central impurity. Within the single excitation subspace the center only couples to the unique symmetric antenna mode (represented by $S^{\dagger } = N^{-1/2} \sum _{j=1}^{N} \hat {\sigma }^{eg}_j$), and $\hat {H}_{\rm eff}$ reads:

Note that the effective field the symmetric ring mode creates at the impurity position corresponds to a single dipole with dipole moment $\sqrt {N} \wp$. Hence the system is formally equivalent to two coupled emitters of unequal dipole moments. The effective ring dipole is detuned from the impurity by $J_R +\delta _I$. A study of this equivalent toy system is given in Supplement 1, Ref. [78].

For a single excitation, the collective eigenmodes of the ring-impurity system and the corresponding decay rates and frequency shifts can be obtained by diagonalizing the $2\times 2$ matrix resulting from projecting Eq. (3) into the subspace spanned by $\left \{ \mathinner {|{R}\rangle }, \mathinner {|{I}\rangle } \right \}$, with $\mathinner {|{R}\rangle } = S^{\dagger } \mathinner {|{g}\rangle }$ and $\mathinner {|{I}\rangle } = \hat {\sigma }^{eg}_I \mathinner {|{g}\rangle }$. Figure 2 shows the decay rate $\Gamma _{\rm min}$ and impurity occupation weight $\mathinner {\langle {\hat {\sigma }^{ee}_I}\rangle }$ respectively, for the most subradiant eigenmode in the case where ring and impurity emitters are identical ($\delta _I = 0$ and $\Gamma _I = \Gamma _0$), as a function of emitter number $N$ and ring size. As central result of this work we find that in the sub-wavelength regime ($\lambda _0 / d \gtrsim 5$) an extremely dark mode with suppressed decay rate $\Gamma _{\rm min}/\Gamma _0 \lesssim 10^{-3}$ emerges exclusively when the ring contains exactly $N=9$ emitters.

 

Fig. 2. Eigenstate properties of the coupled ring-impurity system. (a) Collective decay rate $\Gamma _{\rm min}$ (in units of $\Gamma _0$) and (b) impurity excited state population $\mathinner {\langle {\hat {\sigma }^{ee}_I}\rangle }$, of the most subradiant state emerging in the coupled ring-impurity system, plotted versus $N$ and $\lambda /d$. The figure shows that at sufficiently large value of $\lambda /d$ a very subradiant state for $N=9$ ring emitters exists, whose impurity excited state population is large.

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Its appearance can be understood in the two effective dipoles model, as a subradiant state arises when two dipoles have similar magnitude but opposite phase (singlet configuration) so that their radiated far fields cancel. For a generic state of the form $\mathinner {|{\Psi }\rangle } = \alpha \mathinner {|{R}\rangle } + \beta \mathinner {|{I}\rangle }$ this implies $\beta \approx -\alpha \sqrt {N/\Gamma _I}$. In general, however, such a state is not an energy eigenmode. In the sub-wavelength regime ($\lambda /R \gg 1$), where $\Gamma _R \approx N\Gamma _0$ and $\Gamma \approx \Gamma _0$, this state is only an eigenmode if:

For identical emitters ($\delta _I = 0$ and $\Gamma _I=\Gamma _0$) this reduces to $J_R \approx (N-1)J$. Hence all emitters including the central one experience virtually the same total interaction strength with all others. Indeed we find that the closest integer value $N$ satisfying this condition is $N=9$ as for the most subradiant mode. This reflects a special geometric property of the nonagon (regular polygon with $N=9$ sides), where the sum of the inverse cubic distances $1/r_{ij}^{3}$ to all other N-1 corners is closest to $(N-1)/R^{3}$, i.e., the scaling of the near field dipole-dipole interaction.

Based on this general principle it is straightforward to induce a similar dark mode for other values of $N$ by suitable tuning of the impurity parameters $\delta _I$ or $\Gamma _I$ to fulfill Eq. (5). At small values of $\lambda /d$ the system properties are very sensitive to $\delta _I$ and $\Gamma _I$ and this estimate only yields almost optimal values. In general, the effective polarization strengths $\sqrt {N}\alpha$ and $\sqrt {\Gamma _I} \beta$ associated with the ring and impurity components of the eigenmode are complex values and a minimal decay rate requires a very small imaginary part and a relative phase close to $\pi$. This ensures that short range interactions between the two dipoles do not contribute to the free space energy loss. We show in Fig. 3 the minimum collective decay rate of the coupled system corresponding to $\Gamma _I = \Gamma _0$ when optimizing over the detuning $\delta _I$, as a function of $N$ and $\lambda /d$.

 

Fig. 3. Decay rate (in units of $\Gamma _0$ and in log-scale) of the most subradiant mode of the coupled ring-impurity system, as a function of $N$ and $\lambda /d$, when optimized over the impurity detuning $\delta _I$ (see also Fig. 6 for comparison with the corresponding cross-section).

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4. Absorption cross section

Let us now study the light absorption of this special dark resonance by adding an incoherent decay at rate $\Gamma _T$ from the excited state of the center to an auxiliary state $\left |t\right\rangle$ (see Fig. 1). This implements a sink extracting energy without back-action and could represent irreversible conversion of photons into chemical energy at the reaction center. In an atom based setup extra atoms coupled to the central atom via a dipole moment component orthogonal to the antenna dipoles could extract excitations [79]. For superconducting q-bits one could use a tiny antenna close to the center q-bit [41,80]. Mathematically we simply add a loss term $\mathcal {L}_T [\rho ] = \Gamma _T \left [ \hat {\sigma }^{te}_I \rho \hat {\sigma }^{et}_I - (1/2) \left \{ \hat {\sigma }^{ee}_I,\rho \right \} \right ]$ in the master equation.

Absorption efficiency is quantified by the cross section $\sigma _{\rm abs}$, which represents the effective area for which an incident photon triggers an absorption event. The relative rate of absorbed versus incident photons per area $A$ is then $\sigma _{\rm abs}/A = dn_{\rm abs} / dn_{\rm in}$. In contrast to total scattering or light extinction often used (e.g. [54,81]), our definition of $\sigma _{\rm abs}$ includes both, the probability of scattering a photon by the system and its subsequent transfer to the auxiliary impurity state. Note that the resulting cross section here can exceed the resonant single emitter scattering cross-section $\sigma = 6\pi / k_0^{2}$.

The rate of effectively absorbed photons is $dn_{\rm abs}/dt = \Gamma _T \mathinner {\langle {\hat {\sigma }^{ee}_I}\rangle }$. When all emitters are driven by a perpendicularly propagating coherent field of frequency $\omega _L$ (detuning $\delta =\omega _L-\omega _0$), whose Rabi frequency is $\Omega ({{\mathbf r}}_i) = \Omega$, we add in Eq. (1) the term $\hat {H}_{\rm in} = -\delta \sum _i\hat {\sigma }^{ee}_i+ \sum _i \left [\Omega ({{\mathbf r}}_i) \hat {\sigma }^{eg}_i + h.c.\right ]$, where the index $i$ runs over all dipoles. For a coherent homogeneous drive the incident photon rate over an area $A$ is given by $dn_{\rm in}/dt = 4\Omega ^{2} k_0^{2} A /6\pi \Gamma _0= (4\Omega ^{2}/\Gamma _0) (A / \sigma )$, leading to $\sigma _{\rm abs} /\sigma = \Gamma _T \Gamma _0 \mathinner {\langle {\hat {\sigma }^{ee}_I}\rangle } / 4\Omega ^{2}$. Here the absorption efficiency is compared to a single emitter weakly driven on resonance with spontaneous emission rate $\Gamma _0$ including the additional decay channel at rate $\Gamma _T$. In steady state we have: $\sigma _{\rm abs,\ c}^{\rm single} = \sigma \Gamma _0 \Gamma _T / (\Gamma _0+\Gamma _T)^{2}$, i.e. the product of the probabilities for first scattering a photon and subsequently absorbing it, with maximum value $\max (\sigma _{\rm abs,\ c}^{\rm single} ) = \sigma / 4$ for $\Gamma _T = \Gamma _0$.

For a very low-intensity coherent field we get:

It surpasses a single atom if $\Gamma _{\nu _0}/2 < \left |\langle {I}|{\nu _0}\rangle \right |\cdot \left |\langle {\nu _0^{T}}|{\Omega }\rangle \right |/\Omega$, i.e. the decay rate of the eigenmode has to be small but contain a large impurity occupation weight to compensate for the smaller overlap with incoming radiation. In Fig. 4(a) we show for $N=9$ (identical emitters case) as a function of $\lambda /d$ and detuning $\delta =\omega _L-\omega _0$ of the external coherent drive, the absorption cross section $\sigma _{\rm abs}$ in units of $\max (\sigma _{\rm abs,\ c}^{\rm single} ) = \sigma /4$. This shows that a narrow resonance where the absorption is greatly enhanced emerges for $\lambda /d \gtrsim 5$, exactly corresponding to the frequency of the dark mode previously discussed. In Fig. 4(b) we then plot $\sigma _{\rm abs}$ versus $\lambda /d$ and $N$, when the external drive detuning is tuned to the dark mode collective frequency. In the deep sub-wavelength regime ($\lambda /d \gtrsim 5$) a distinct maximum cross-section arises for $N=9$ emitter antennae and the regions of maximal absorption correspond to those with minimum collective decay.

 

Fig. 4. Absorption cross-section $\sigma _{\rm abs}$ (in units of maximal single atom absorption cross section $\max (\sigma _{\rm abs,c}^{\rm single})= \sigma /4$) of the coupled ring-impurity system illuminated by a weak coherent circularly polarized perpendicularly propagating field. (a) $\sigma _{\rm abs}$ versus external field detuning $\delta$ and $\lambda /d$ for $N=9$. For small enough rings ($\lambda /d \gg 1$) the system is a frequency-selective antenna with resonantly enhanced absorption cross-section, corresponding to the dark eigenmode and which can be tuned via the system parameters. (b) $\sigma _{\rm abs}$ versus $N$ and $\lambda /d$ for resonant light with the subradiant eigenmode. For $\lambda /d \gtrsim 5$ a maximum in the absorption occurs exactly for $N=9$ ring emitters, where the collective mode is most subradiant (see for comparison Fig. 2 (a)).

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As said, a similar dark mode can be accessed for different values of $N$ by tuning the impurity parameters $\Gamma _I$ and $\delta _I$, which yields an enhanced absorption cross section as shown in Fig. 5(a) and (b), where we plot $\sigma _{\rm abs}$ versus $N$ and $\delta _I$ (at fixed $\Gamma _I=\Gamma _0$), or $\Gamma _I$ (at fixed $\delta _I = 0$), for $\lambda /d = 20$. Finally, in Fig. 6 we depict $\sigma _{\rm abs}$ as a function of $N$ and $\lambda /d$, for $\Gamma _I = \Gamma _0$ and the optimal value of $\delta _I$, showing that enhanced absorption $\sigma _{\rm abs}$ with respect to the single emitter case can be achieved for an arbitrary value of $N \geq 3$ by tuning the impurity parameters.

 

Fig. 5. Effect of impurity detuning $\delta _I$ and decay rate $\Gamma _I$ in the coupled ring-impurity system. Absorption cross section $\sigma _{\rm abs}$ (in units of maximal single atom absorption cross section $\max (\sigma _{\rm abs,c}^{\rm single}) = \sigma /4$) versus $N$ and (a) $\delta _I$ at fixed $\Gamma _I = \Gamma _0$, and (b) $\Gamma _I$ at fixed $\delta _I = 0$, for $\lambda _0/d = 20$. The dashed white lines are Eq. (5. Inset: $\sigma _{\rm abs}$ for $N=9$ (red line), dark mode decay rate (log-scale, black line), effective dipole moment $|\wp _{\rm eff}|^{2}$ (cyan line) and $\rm {Im}[\wp _{\rm eff}]$ (blue line, log-scale), for comparison. $\sigma _{\rm abs}$ is very sensitive to the detuning $\delta _I$ with a sharp maximum near the minimum of $\rm {Im}[{\wp _{\rm eff}}]$.

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Fig. 6. Absorption cross-section $\sigma _{\rm abs}$ (in units of maximal single atom absorption cross section $\max (\sigma _{\rm abs,c}^{\rm single}) = \sigma /4$) versus $N$ and $\lambda /d$ optimized as function of $\delta _I$ at fixed $\Gamma _I = \Gamma _0$. The dashed white line represents $\lambda /R=1$. For optimal $\delta _I$ enhanced absorption with respect to the single atom case occurs for $N\geq 3$ at $\lambda /R \geq 1$.

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5. Enhancement of incoherent light absorption

So far we dealt with spatial and temporal coherent input radiation with well defined intensity as it occurs for incoming laser light. Of course this is far from the conditions present for thermal radiation sources, where only spatial and not temporal coherence is present. It is, however, straightforward to generalize our absorption model to account for a spectral bandwidth of the incoming light. Actually in the very large bandwidth limit one simply has to replace the coherent driving amplitude by a temporally incoherent but spatially coherent field. Mathematically this just amounts to add the excitation rates in form of an extra Liouvillian term in the master equation:

 

Fig. 7. Absorption cross-section $\sigma _{\rm abs}$ (in units of single emitter $\sigma _{\rm abs,\ inc}^{\rm single}=\sigma \Gamma _T/(\Gamma _0+\Gamma _T)$, see main text) in the coupled ring-impurity system for an incoherent weak pump. (a) $\sigma _{\rm abs}$ versus $N$ and $\Gamma _T/\Gamma _0$, at fixed $\lambda /d = 40$, and (b) versus $N$ and $\lambda /d$, at fixed $\Gamma _T/\Gamma _0 = 10^{-4}$. An enhancement in absorption with respect to the single emitter case is found for $N=9$ at sufficiently small $\Gamma _T/\Gamma _0$ and large $\lambda /d$.

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6. Semi-classical coupled dipole model

As we have seen above the enhancement is close tied to a very dark collective eigenstate of the system with ample weight on the center dipole. As it has been argued that the most dark states are entangled [12,85], one can ask whether our results hold in the case of classical dipole arrays. To this end we can simply apply a mean-field type of approximation to the quantum description, representing atomic operators by their mean values and study the differences to the quantum model, where we restricted the Hilbert space to the single or the two excitation manifold. In this classical limit we get simple coupled c-number Bloch equations for each dipole and the center impurity. For symmetry reasons the expectation values for all ring atoms follow the same time evolution, so that finally we end up with a rather small finite closed set of differential equations:

 

Fig. 8. Comparison of the absorption cross section $\sigma _{\rm abs}$ (in units of maximal single atom absorption cross section $\max (\sigma _{\rm abs,c}^{\rm single})= \sigma /4$) for a coherent drive obtained with the quantum model (one and two-excitation truncated states as indicated in the legend) and with the classical (mean-field) approximation. (a) $\sigma _{\rm abs}$ as a function $\lambda /d$ for the $N=9$ ring and a coherent pumping rate $\Omega /\Gamma _0 = 5 \times 10^{-4}$ and detuning $\delta = \omega _\textrm {min}$, with $\omega _\textrm {min}$ being the collective frequency of the subradiant mode. The trapping rate is fixed to $\Gamma _T = \Gamma _\textrm {min}$ and all emitters are circularly polarized. (b) $\sigma _{\rm abs}$ for $d = \lambda /12$ and $N=9$, as a function of $\Omega$ and for the same $\delta$ and $\Gamma _T$ as in (a). The lines are only guides to the eye.

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Interestingly, in the case of very close dipoles, larger differences appear and the quantum model always predicts a superior cross section by looking at a cut along the $N=9$ line. Note that for very small diameters $\lambda /d > 8$ in the sub-wavelength region the quantum model predicts a significantly larger absorption cross section. In this regime, the energy shifts get more important than the modifications of the decay properties. One might speculate that this makes coherence more robust, increasing the absorption efficiency.

7. Conclusions

A sub-wavelength regular nonagon of dipoles has a built in geometric symmetry allowing for the existence of unique subradiant eigenstates with a high center population weight, if one adds an additional equivalent centered absorber acting as an energy dump. This special property enhances absorption of light for weak uniform illumination not only in the case of resonant enhancement for spectral narrow radiation but it also appears for broadband incoherent light. Actually the absorption enhancement is even much stronger when only the antenna dipoles are selectively illuminated, while the center impurity is shielded. While such selective illumination seems hard to implement technically, a dynamical mechanism switching the center dipole on and off or shift its frequency should induce a similar effect. Note that comparable enhancements can also be engineered for other antenna sizes and geometries if one optimizes the center impurity strength and resonance frequency.

Interestingly, a classical mean field description reproduces these results well for larger ring dimensions and atom numbers, while the strongest enhancement at sub-wavelength distances appears only in a quantum treatment. Preliminary studies beyond weak field illumination also reveal a suppression of absorption of a second photon as long as the system is in the excited state. This will presumable suppress the g2 intensity correlation function of the emitted fluorescence as has recently also been found in related studies [86,87]

In practise, such dark state nano-ring antenna configurations inspired by natural ring structures should find applications in nanoscale single photon detection or even spectroscopy. Operated in reverse these structures act as coherent light nano-sources [64] or even non-classical single photon sources [87].

Let us finally remark, that while this effect appears not to be directly exploited in single natural LHC2 molecules, already the more complex form of of LHC1 is composed of a ring structure with a reactive center, where the proposed mechanism could be at work and in particular in combined LHC2-LHC1 compounds [54]. As these whole structures are very well below a wavelength in size, even thermal light will couple directly to collective delocalized excitations as starting point of light absorption. Clearly, light to chemical energy conversion in a thermal environment with a complex reaction center is a way more complex process. Nevertheless, it is hard to imagine that the ring symmetry found in many biological realizations is not used [84] and a pure coincidence.