1. Introduction

In quantum optics, ordered quantum emitter lattices with subwavelength spacing have emerged as a resourceful platform for near-term quantum technologies [1–12]. Here long-range interactions between light-induced dipoles lead to highly modified optical properties of the quantum emitter ensemble, including Dicke superradiance [13] and the emergence of collective long-lived subradiant states [14,15]. Applications range from single-photon switch gates [16] to enhanced single-photon detection for biomedical applications [17,18] and topological edge state lasing [19,20]. Likewise, uncovering design principles underlying biological systems and applying this understanding to synthetic systems is crucial for near-term quantum technologies. Ring geometries of quantum emitters promise to enhance single-photon sensing, transport, storage, and light generation in engineered nanoscale systems [15,21–23]. In photosynthetic energy transfer, as it occurs in nature, organisms use ring-shaped antennae that increase the photon scattering cross section of a single reaction center: the site where photosynthesis takes place. This transfer process occurs at near unit efficiency, and understanding the mechanisms behind this remarkable feat is an outstanding scientific challenge [24–34].

Taking inspiration from biological systems, we examine the long-range excitation transport between a donor and an acceptor emitter through a lattice of quantum emitter rings [Fig. 1(a)]. As a main result, we show that efficient excitation transport at low trapping rates preferentially occurs for ring geometries, as compared with other lattices. This property has important consequences for devising artificial light harvesting and transport systems, and may be relevant for understanding the excellent excitation transport capabilities of biological systems [35]. We also highlight that for ring lattices, the trapping of light at an acceptor site under low-light conditions is enhanced by many orders of magnitude as compared to other geometries and to independent emitters. By choosing an optimal detuning for the donor and acceptor with respect to the lattice, radiative losses are strongly suppressed, and excitations are protected during the transport by subradiance [14,15]. While the influence of the excitation trapping rate on the transport efficiency in other geometries has been explored in other works [36–39], as have certain design principles for bio-inspired artificial solar-harvesting devices [24,26,27,40–44], our findings specifically highlight the advantages of the rotationally symmetric ring geometry. This special feature of ring configurations is particularly intriguing for its close connection to natural photosynthetic complexes found in biological systems. Our work therefore opens the possibility of exploiting quantum effects in bio-inspired configurations of quantum emitters for near-term optical technologies that enable quantum-enhanced light–matter coupling on the nanoscale.

 

Fig. 1. Lattices of nanoscopic quantum emitter rings. (a) Each ring is composed of two-level quantum emitters with resonance frequency $\omega _0$ and separation $d<\lambda _0$, where $\lambda _0=\omega _0/c$ is the wavelength of light. The excited state $|e\rangle$ spontaneously decays with rate $\Gamma _0$ to the ground state $|g\rangle$ and the emitters are coupled via long-range dipole–dipole interactions with nearest-neighbor coupling strength $J$. Emitters acting as donor and acceptor are shown in yellow and the acceptor features an additional trapping state to which excitations irreversibly decay with rate $\Gamma _\mathrm {T}$. (b) More detailed sketch illustrating the inter-ring separation $d_R$. The ring radius $R$ and the emitter spacing $d$ are related via $d = 2R \sin (\pi /N_\mathrm {R})$, with $N_R$ emitters per ring. (c) Excitation transport efficiency according to Eq. (1) for a chain of 10 rings and various $N_R$. Parameters: $d/\lambda _0 = 0.05$, $d_R/d=0.9$, $\Gamma _\mathrm {T}/\Gamma _0 = 2$, and $\Delta = 0$.

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2. Quantum Optical Model

As a paradigmatic quantum optical model to simulate excitation energy transport, we consider a one-dimensional lattice of $M$ rotationally symmetric rings, each composed of $N_R$ identical two-level emitters with a ground state $|g \rangle$ and an excited state $|e\rangle$. The two states are connected via the transition operator $\hat {\sigma }_n = |g_n \rangle \langle e_n|$ for the $n$th emitter. Additional emitters acting as donor and acceptor sites are placed in the center of two rings at either end of the lattice, as illustrated in Fig. 1(a). The transport efficiency between these two sites is the core quantity of interest and is defined as

We model the system within the Born–Markov approximation [14], and only consider the quantum emitter’s internal degrees of freedom. Furthermore, we assume the weak excitation regime, where at most a single excitation is present in the system (see Supplement 1 for details), and therefore the system can be described (in the rotating frame with $\omega _0$) by the non-Hermitian Hamiltonian $\hat {\mathcal {H}}_\mathrm {eff} = \hat {\mathcal {H}}_\mathrm {ad}+\hat {\mathcal {H}}_\mathrm {lattice}+\hat {\mathcal {H}}_\mathrm {int}$. Here, $\hat {\mathcal {H}}_\mathrm {ad} = (\Delta -\frac {i}{2}\Gamma _0) (\hat {\sigma }^\dagger _\mathrm {a}\hat {\sigma }_\mathrm {a}+\hat {\sigma }^\dagger _\mathrm {d} \hat {\sigma }_\mathrm {d}) - \frac {i}{2}\Gamma _\mathrm {T} \hat {\sigma }^\dagger _\mathrm {a}\hat {\sigma }_\mathrm {a}$ is the bare Hamiltonian of the donor and acceptor, $\hat {\mathcal {H}}_\mathrm {lattice}$ describes the emitters in the ring lattice, and $\hat {\mathcal {H}}_\mathrm {int}$ describes the interaction between the ring emitters and the donor/acceptor,

All emitters interact via vacuum-mediated dipole–dipole interactions in free space. The pairwise coherent and dissipative interactions are given by $J_{nm} = -3\pi \Gamma _0 /k_0 \ \mathrm {Real} (G_{nm})$ and $\Gamma _{nm}=6 \pi \Gamma _0 /k_0 \ \mathrm {Imag} (G_{nm})$, respectively, with $G_{nm}$ being the free space Green’s function (see Supplement 1 for details). The Green’s function depends only on the separations between the emitters and their dipole orientation. The time evolution of the system is described by the effective Hamiltonian in Eq. (2) via the Schrödinger equation $i \partial _t | \Psi (t) \rangle = \hat {\mathcal {H}}_\mathrm {eff}| \Psi (t) \rangle$. Since the Hamiltonian is non-Hermitian, the amplitude of the wavefunction for the quantum emitters decreases with time, which is a direct manifestation of the dissipative nature of the system.

3. Collective Modes

As discussed in previous works [14,15,21], a subwavelength-spaced ring of quantum emitters exhibits guided eigenmodes that are extremely subradiant, exhibiting an exponentially increasing lifetime, $\tau \Gamma _0 \sim \mathrm {exp}(N_R)$, of a single excitation [21]. Aside from the bright symmetric superposition state, the fields of the remaining eigenmodes vanish at the center of the ring due to symmetry. Thus, they are decoupled from any emitter at the center. Here we demonstrate that a donor/acceptor at the center of the ring that is dipole–dipole coupled to the symmetric ring mode can form a subradiant state with a majority of the excitation concentrated in the donor/acceptor. We start by analyzing a single ring of $N_R$ emitters with a single donor in the center. For a single ring, where the dipole orientations preserve the discrete rotational invariance, the collective eigenmodes of the effective Hamiltonian are spin waves of the form $|\Psi _m \rangle = \hat {S}_m^\dagger |G\rangle$, where

Likewise, a chain of quantum emitter rings features a rich collective eigenmode structure. In particular, the subradiance of the eigenmodes protects the excitations from radiative decoherence and leads to efficient excitation transport [15,45]. As shown above, eigenmodes of a rotationally symmetric ring carry angular momentum $m$. Similarly, the eigenmodes of a linear chain of quantum emitters carry linear momentum $k$ [5,14]. This leads to an ansatz wavefunction for the eigenmodes of a ring chain, $|\Psi _{m,k} \rangle$, with an angular and linear quasi-momentum pair $(m,k)$, and associated eigenenergies $\omega _0+J_\mathrm {m,k}$ and decay rates $\Gamma _\mathrm {m,k}$ [46] (details provided in Supplement 1). The translational distance between adjacent ring centers along the chain is given by $\tilde {d}=2R+d_R$. Figure 2(a) shows the energy bands for $N_R=9$ and $d_R/d = 0.9$. The band structure exhibits a nontrivial topology with a non-zero Zak phase $\varphi = i \int _{BZ} d k \, \langle \Psi _{m=0,k} | \partial _k| \Psi _{m=0,k} \rangle$ [47] as well as gapped edge states between the energy bands of the $m=0$ and $|m|=1$ eigenmodes. The edge states emerge for decreasing inter-ring spacings $d_R$, illustrated in Fig. 2(c), and become more pronounced until a critical spacing of $d_R/d\approx 0.58$ and $d_R/d \approx 0.34$ for $N_R=8,9$ emitters per ring, respectively. The edge states are energetically degenerate and detuned by $\Delta _\mathrm {gap}^{(0,1)}$ from the lower/upper band edge, respectively. Figure 2(b) shows the minimum distance of the edge states to the nearest band edge as a function of the inter-ring spacing $d_R$. Topologically protected edge states are crucial for resilient excitation transport in disordered systems [4] and $\mathrm {min}(\Delta _\mathrm {gap}^{(0,1)})$ can serve as a figure of merit in this regard. Specifically for lattices of rings, the band gap remains finite in the presence of rotational disorder and exhibits a distinct maximum, e.g., $d_R/d \approx 0.9$ for $N_R=9$, as shown in Fig. 2(b). Edge states also become superradiant at the critical distance where excitation transport is surpressed, as is shown in Fig. 2(c). This points to the possibility of edge mode lasing, already observed in gold nano-rings arranged in zigzag chains [19,20,48].

 

Fig. 2. Band structure, edge states, and excitation transport for a chain of rings. (a) Eigenmodes of the effective Hamiltonian $\hat {\mathcal {H}}_\mathrm {lattice}$ in Eq. (2a) can be cast into $N_R=9$ energy bands with an angular momentum projection $|m|$ using the Ansatz shown in Supplement 1 and translation along the ring chain axis is given by $\tilde {d}=2R+d_R$, the spacing between adjacent ring centers. (Decay rates of all eigenmodes are color coded.) A band gap emerges with two edge states residing inside with an energy separation $\Delta _\mathrm {gap}^{(0,1)}$ to the nearest lower/upper band edge respectively. (b) Minimal energy gap $\Delta _\mathrm {gap}^{(0,1)}$ to the nearest lower/upper band edge normalized by the maximal nearest-neighbor coupling $J$ shows distinct maxima as a function of $d_R$. These maxima correspond to an optimal transport efficiency $\eta _t$ as shown in panel (e) for $N_R=8,9$. Furthermore, smaller emitter numbers per ring $N_R$ are affected more by randomly rotated rings except for even $N_R$. The average was taken over 50 random realizations and where the rings can be randomly rotated between $\pm 2\pi /N_R$. (c) For $N_R=9$, edge states appear with maximum amplitude on the left/right end of the ring chain and a superradiant decay rate for decreasing inter-ring spacings $d_R/d$. Notably, topological edge states have been observed in zigzag chains of gold nano-rings [19,20] with lasing from the edge rings. (d) With decreasing inter-ring spacing $d_R$, edge states become more pronounced with distinct minima where the bulk amplitude vanishes leading to a suppression of excitation transport. (e) Excitation transport between a donor and acceptor site placed in the center of the edge rings. The transport efficiency $\eta _t$ is evaluated after a time $t \Gamma _0 = 150$ with the donor/acceptor detuning $\Delta$ optimized for maximum transport. Suppression appears when the edge states become too pronounced with vanishing amplitude in the bulk rings as shown in panel (d). (f) Time dynamics of the excitation transport process between a donor and acceptor site with $\Delta =0$. The eigenstate fidelity $|\langle \Psi (t)| \Psi _\mathrm {eig} \rangle |^2$ for $N_R=9$ demonstrates the importance of edge states at early times. Parameters: $d=0.05\lambda _0$, $\Gamma _\mathrm {T}/\Gamma _0 = 1$ in panels (e),(f) and 10 rings in panels (b)–(f).

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Figures 2(e) and 2(f) demonstrate the fundamental influence of the edge states on the transport dynamics between a donor and acceptor site for 10 rings with $N_R=9$. At the critical spacings, transport is either completely or strongly suppressed because the edge states possess no amplitudes in the bulk rings. Conversely, at other spacings $d_R$, edge states are crucial during the early times of the transport process, as demonstrated by the eigenstate fidelities $\mathcal {F}(t) = |\langle \Psi (t)| \Psi _\mathrm {eig} \rangle |^2$ for $N_R=9$. Qualitatively similar results hold for other $N_R$. Indeed, edge states have been thoroughly studied in dimerized chains ($N_R=2$), which reproduces a long-range generalization of the well-known Su–Schrieffer–Heeger (SSH) model [20]. A more complete discussion of the emergence of edge and corner states [48] in two-dimensional ring lattices is briefly discussed in Supplement 1 and warrants further study.

4. Results: Excitation Transport and Trapping

We now focus on the excitation transport dynamics and discuss the time evolution of a single initially excited donor. We find that excitation transport is optimized at particular donor/acceptor detunings, and that efficient transport occurs only for ring emitter numbers $N_R \ge 6$. In particular, rings with 8-, 9-, and 10-fold symmetry seem to be most optimal. This is particularly intriguing because 8-, 9-, and 10-fold rings, the most abundant type occurring in natural light harvesting antennae, show the highest resilience when rings are randomly rotated with respect to each other [49–51]. In Fig. 3, the donor excited state populations and the trap populations $\eta _t$ are shown after a time $t \Gamma _0 = 150$ for a chain of 10 rings with various inter-ring spacings $d_R$. The detuning $\Delta$ where the donor excited state population is maximized (i.e., most subradiant) follows the optimal detuning $\Delta _\mathrm {sub}$ for the single-ring case. Here, the donor excitation largely remains trapped in the subradiant state discussed above, even for small inter-ring spacings.

 

Fig. 3. (a)–(c) Excitation transport in quantum emitter rings between a donor and acceptor site. Scan over the number of emitters per ring $N_R$ and donor/acceptor detuning $\Delta$ for 10 rings with decreasing inter-ring spacings $d_R$ after a time $t \Gamma _0 = 150$. Efficient transport emerges only with $N_R \ge 6$, irrespective of $d_R$. The excited state population in the donor can get trapped in a subradiant state involving the ring surrounding the donor and follows the detuning $\Delta _\mathrm {sub}$ (white dashed line) derived in the main text. For 9-fold symmetric rings, the donor/acceptor detuning that optimizes transport is given by $\Delta \approx 0$ for all inter-ring spacings. Additional parameters are: $\Gamma _\mathrm {T} = 2 \Gamma _0$, $d=0.05\lambda _0$.

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A crucial element of excitation energy transport is robustness against energy disorder. We provide a comparison of ring lattices with other lattice geometries, including the influence of static frequency disorder in the lattice emitters. This is achieved by taking emitter frequencies $\omega _m$ from a Gaussian distribution around the unperturbed emitter frequency $\omega _0$ with a standard deviation $\delta \omega$ and adding the term $\sum _m (\omega _m-\omega _0) \hat {\sigma }^\dagger _m \hat {\sigma }_m$ to the Hamiltonian in Eq. (2). The donor/acceptor detuning $\Delta$ remains unchanged and is chosen such that unperturbed excitation transport is maximized. Figure 4 shows various geometries, many of which also have been studied previously for atoms trapped in optical lattices [4,5,10,12,21,45]. The nearest-neighbor distance is kept at $d=0.06 \lambda _0$ and the donor–acceptor distance at $\sim \lambda _0$ to establish a uniform comparison between the different geometries. Figures 4(a) and 4(b) show a hexagonal lattice with $\Delta =0$ and a honeycomb lattice with $\Delta = 4.5 \ \Gamma _0$. In Fig. 4(c), a 1D ring chain and 2D hexagonal ring lattice with $N_R=9$ are shown with $\Delta = \Gamma _0$. The fluctuation in the emitter frequencies $\delta \omega$ is set to $|J|/4$ and $|J|/2$, where $J \approx -8.4 \Gamma _0$. Altogether, the different geometries show a similar reduction in the maximal transport efficiencies under disorder but behave quite differently in the range of trapping frequencies $\Gamma _\mathrm {T}$ where maximal transport occurs. Whereas the hexagonal lattice exhibits peak transport at a trapping rate far above the optical decay rate, namely at $\Gamma _\mathrm {T}/|J| \sim 2$, the ring lattices demonstrate efficient transport over a large range $\Gamma _\mathrm {T}^\mathrm {opt}/|J| \sim 0.01\text {--}1$, even in the moderately disordered case. Qualitatively similar conclusions also apply to ring lattices with $N_R \neq 9$. Just as importantly, the ring lattices in Fig. 4(c) show significant transport enhancement for $\Gamma _\mathrm {T} \ll |J|$ compared to the independent case where no lattice is present. In summary, the ring lattices show significantly better transport capability and robustness against disorder.

 

Fig. 4. Excitation transport in ring geometries exhibits superior robustness against disorder. Comparison of transport efficiency in (a) chain and hexagonal lattices and in the absence of a lattice (gray line), (b) honeycomb, and (c) ring lattices as a function of the trapping rate $\Gamma _\mathrm {T}$ and frequency disorder. Lattice emitter frequencies are randomly fluctuating by $\delta \omega$ around the resonance frequency $\omega _0$. The donor–acceptor distance is approximately the wavelength of light $\lambda _0$. Also shown in the dash-dotted lines is the case of a donor–acceptor pair separated by $d$ in the absence of any lattice. Although frequency disorder decreases the long-range transport capacity, it prevails remarkably well even at large frequency fluctuations. In particular, the ring lattices exhibit high transport efficiencies (close to 90${\% }$) over a wide range of trapping rates as compared to the other geometries. At trapping rates much below the magnitude of the coherent transfer rate $J$, ring-based lattices are superior to any other lattice in our study. Additional parameters: $d_R/d= 0.9$, $d/\lambda _0 = 0.06$, $J/\Gamma _0 \approx -8.4$, $t\Gamma _0=150$. An average over 25 random realizations with standard deviation $\delta \omega$ was performed in all plots. Donor/acceptor detunings of (a) $\Delta = 0$, (b) $\Delta = 4.5 \Gamma _0$, and (c) $\Delta =-\Gamma _0$.

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So far, we have assumed that a single donor is initially excited, and we have quantified the transport behavior by calculating the fraction of the excitation that accumulates in a trap state via Eq. (1) after a waiting time $t$. However, in many realistic scenarios, a perfectly excited donor is rather unlikely, and emitters close to the donor will be excited too. This motivates the study of the trapping rate at which the excitation ends up in the trap state under continuous coherent illumination in the form of a Gaussian laser beam with finite beam waist $w$. The continuous coherent drive is modeled by

 

Fig. 5. Ring lattices exhibit the most efficient energy trapping under weak light illumination and small trapping rates. A comparison of the steady-state effective trapping rate $\Gamma _\mathrm {T} \langle \hat {\sigma }_a^\dagger \hat {\sigma }_a \rangle _\mathrm {st} / (4\Omega _0^2)$ between different lattice geometries under continuous coherent driving with rate $\Omega _0 / \Gamma _0 = 10^{-3}$. The drive is modeled by a Gaussian beam with waist $w$ centered at the acceptor emitter and is on-resonance with the lattice emitter frequencies $\omega _0$ [see Eq. (4)]. The effective trapping rate is normalized by the single emitter trapping rate driven on resonance (see main text). We study a hexagonal and chain lattice in panel (a) with $20$ and $8 \times 9$ emitters, a honeycomb lattice in panel (b) with 130 lattice emitters, a chain of five rings in panel (c), and a $3 \times 3$ hexagonal ring lattice in panel (d). The donor/acceptor distance is approximately the wavelength of light $\lambda _0$ in all lattices. Strikingly, only the ring-based lattices are orders of magnitude more efficient at trapping incoming light for trapping rates below the nearest-neighbor coupling rate, namely when $\Gamma _\mathrm {T} \ll |J|$. Conversely, the honeycomb and hexagonal lattices are orders of magnitude less efficient in the same regime with the honeycomb lattice not deviating significantly from the single acceptor case. The linear chain (gray dashed) and the free space transfer (solid gray) shown in panel (a) is more efficient at trapping rates $\Gamma _\mathrm {T} > |J|$. Additional parameters: $d/\lambda _0 = 0.06$, $J/\Gamma _0 \approx -8.4$ and donor/acceptor detunings of (a) $\Delta /\Gamma _0 = -18$ (hexagonal), $\Delta = 0$ (chain), (b) $\Delta /\Gamma _0 = -20$, (c) $\Delta /\Gamma _0 = -3.85$, (d) $\Delta /\Gamma _0 =-4.63$ with $d_R/d = 0.9, \ N_R=9$, in panels (c) and (d).

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5. Conclusions

In conclusion, we have demonstrated intriguing optical properties of quantum emitter ring lattices, including the emergence of topological edge states. Furthermore, we have shown based on general symmetry principles that ring lattices form a superior platform for transporting and trapping excitations. We have also elucidated the guiding principles that govern optimal donor/acceptor detunings, trapping rates, and geometric arrangements with robustness against static energy disorder. Under more realistic conditions of weak coherent light illumination, we have shown that ring lattices are orders of magnitude more efficient at trapping the absorbed light when the trapping rate is much smaller than the nearest-neighbor coherent coupling rate. This result is thought-provoking since natural light-harvesting systems also operate with trapping mechanisms that are orders of magnitude slower than the coherent transfer time between neighboring chromophores [52,53]. Measurements performed on natural light-harvesting complexes show that the coherent energy transfer time between neighboring chromophores during photosynthesis is of the order of $\sim 0.1\text {--}10$ ps whereas the trapping time in the reaction center is typically $\sim 0.1\text {--}10$ ns [35,52,53]. For a pre-existing trapping structure, this could provide an explanation why nature uses ring geometries as a moderating mechanism to trap absorbed sun light in reaction centers. Other studies have focused on molecular emitters in ambient conditions with vibrational degrees of freedom as well as multiple decoherence channels [54–56]. These works have shown that coupling between electronic and vibronic modes can aid the transport and trapping processes in ordered molecular arrays such as ring geometries via unidirectional transfer from superradiant to subradiant collective electronic states [57–59]. The impact of these additional effects on the results presented here are an exciting avenue for future research [60–62]. Nevertheless, our results suggest that there exist general and platform-agnostic design principles that govern the efficient transport of excitation energy at the nano scale. These geometrical considerations may have played a role in evolutionary design and warrant further study.