INTRODUCTION

In quantum optics, ordered quantum emitter lattices with subwavelength spacing have emerged as a resourceful platform for near-term quantum technologies [1][2][3][4][5][6][7][8][9][10][11][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][22][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][25][26][27][28][29][30][31][32][33][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][37][38][39], as have certain design principles for bio-inspired artificial solar-harvesting devices [24,26,27,[40][41][42][43][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.

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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⟩ and an excited state |e⟩. The two states are connected via the transition operator σn = |g n ⟩⟨e n | for the nth 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

Here t is the integration time over which excitation can accumulate in the trap state [see Fig. 1(a)], η t can take values between 0 and 1, where 0 corresponds to no transport at all and 1 identifies maximal transport efficiency, and σ † a σa = |e a ⟩⟨e a | corresponds to the projector onto the excited state of the acceptor. The trap population accumulates over time and reaches a steady-state value at large times t when the total excited state population is either dissipated via radiative losses or accumulated in the trap. The transition frequencies and decay rates of the ring emitters are assumed to be equal and given by ω 0 = 2πc/λ 0 and Γ 0 , respectively, whereas the donor/acceptor transitions may be detuned by ∆ = ω d,a -ω 0 with respect to the ring emitter frequencies. We assume ω d = ω a for the remainder of this work. The acceptor features an extra trapping channel through which excitations are extracted from the system at a rate Γ T . Furthermore, the quantum emitters are confined in the x-y plane with intra-ring separation d = 2R sin(π/N R ) and inter-ring separation d R , where R is the ring radius, as illustrated in Fig. 1(b). To reduce the number of free parameters, all dipole emitters are assumed to be circular polarized, namely (1, i, 0) T / √ 2. However, qualitatively similar results can be obtained for geometries consisting of linear polarized emitters.

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 ω 0 ) by the non-Hermitian Hamiltonian Ĥeff = Ĥad + Ĥlattice + Ĥint . Here,

a σa is the bare Hamiltonian of the donor and acceptor, Ĥlattice describes the emitters in the ring lattice, and Ĥint 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 = -3πΓ 0 /k 0 Real(G nm ) and Γ nm = 6πΓ 0 /k 0 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∂ t |Ψ(t)⟩ = Ĥeff |Ψ(t)⟩. 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.

COLLECTIVE MODES

As discussed in previous works [14,15,21], a subwavelengthspaced ring of quantum emitters exhibits guided eigenmodes that are extremely subradiant, exhibiting an exponentially increasing lifetime, τΓ 0 ∼ 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 and |G⟩ denotes all emitters in the ground state. Here φ j = 2πj/N R is the angle between neighboring emitters along the ring and m = 0, ±1, • • •, ⌈±(N R -1)/2⌉ is the angular momentum of the collective mode. The associated energy shifts and decay rates of these spin waves are given by Jm = ∑︁ j e imϕ j J 1j and Γm = ∑︁ j e imϕ j Γ 1j , respectively. In such a configuration, all ring emitters couple equally to the central donor, which restricts the spectrum to the m = 0 mode. This system features two eigenstates |Ψ ± ⟩ that are symmetric/anti-symmetric superpositions of the symmetric ring mode and the central donor. The anti-symmetric state can be extremely subradiant depending on the detuning ∆ of the donor with respect to the ring emitters, resulting in a vanishingly small net dipole strength [23]. This leads to an optimal detuning ∆ sub ≈ J d ( Γ0 -Γ 0 ) -J0 that maximizes the subradiance of the donor with an effective decay rate Γ eff /Γ 0 ≲ 10 -3 (see Supplement 1 for details). Here, J d is the coherent coupling between the donor and a ring emitter.

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, |Ψ m,k ⟩, with an angular and linear quasi-momentum pair (m, k), and associated eigenenergies ω 0 + J m,k and decay rates Γ m,k [46] (details provided in Supplement 1). The translational distance between adjacent ring centers along the chain is given by d = 2R + d R . Figure 2 gap 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 min(∆ (0, 1) gap ) 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 ≈ 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].

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,

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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 F (t) = |⟨Ψ(t)|Ψ eig ⟩| 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.

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 ≥ 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][50][51]. In Fig. 3, the donor excited state populations and the trap populations η t are shown after a time tΓ 0 = 150 for a chain of 10 rings with various inter-ring spacings d R . The detuning ∆ where the donor excited state population is maximized (i.e., most subradiant) follows the optimal detuning ∆ 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. 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 ω m from a Gaussian distribution around the unperturbed emitter frequency ω 0 with a standard deviation δω and adding the term ∑︁ m (ω m -ω 0 ) σ † m σm to the Hamiltonian in Eq. ( 2). The donor/acceptor detuning ∆ 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 = 0.06λ 0 and the donor-acceptor distance at ∼ λ 0 to establish a uniform comparison between the different geometries. Figures 4(a , where J ≈ -8.4Γ 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 Γ T where maximal transport occurs. Whereas the hexagonal lattice exhibits peak transport at a trapping rate far above the optical decay rate, namely at Γ T /|J| ∼ 2, the ring lattices demonstrate efficient transport over a large range Γ opt T /|J| ∼ 0.01-1, even in the moderately disordered case. Qualitatively similar conclusions also apply to ring lattices with N R ≠ 9. Just as importantly, the ring lattices in Fig. 4(c) show significant transport enhancement for Γ T ≪ |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.

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

where Ω 0 is the laser Rabi frequency and ⃗ r d is the position of the donor with the sum including all emitters. The driving rate of the laser is kept small (Ω 0 ≪ Γ 0 ) to ensure that the system stays in the single-excitation regime and the model remains valid. As a figure of merit for energy transport efficiency, we define Γ T ⟨ σ † a σa ⟩ st /(4Ω 2 0 ) as the steady-state trapping rate at the acceptor emitter. The effective trapping rate is normalized by the trapping rate of a single acceptor, given by σ 0 Γ 0 Γ T /(Γ 0 + Γ T ) 2 , where σ 0 = 6π/k 2 0 is the single emitter scattering cross section [23]. In Figs. 5(a)-5(c), a hexagonal lattice, a honeycomb lattice, and a hexagonal ring lattice with N R = 9, respectively, are compared for two laser beam waists under continuous driving. By choosing a beam waist of w/λ 0 = 0.3, most of the incoming light is focused around the donor emitter while the acceptor emitter remains mostly undriven. For w/λ 0 = 3, the whole lattice is uniformly driven-a scenario more applicable to deeply subwavelength lattices under illumination from a non-directional light source. Natural light-harvesting antennae in purple bacteria offer an example [24]. In all cases, the donor/acceptor detuning ∆ is chosen optimally such that the trapping rate is maximized. We find that the ring lattice is many orders of magnitude more efficient in trapping incident light as compared to both the triangular and honeycomb lattices as well as the single emitter at trapping rates much below the nearest-neighbor coherent transfer rate J. In particular, for Γ T /|J| ≲ 0.01, the ring lattice exhibits an almost 100× higher trapping efficiency compared to an independent emitter and the other lattices.

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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 ∼ 0.1-10 ps whereas the trapping time in the reaction center is typically ∼ 0.1-10 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][55][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][58][59]. The impact of these additional effects on the results presented here are an exciting avenue for future research [60][61][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.

Funding. Austrian Science Fund (DK-ALM W1259-N27); Harvard Quantum Initiative (HQI); AFOSR and the NSF (through the CUA PFC and QSense QLCI).

Disclosures. The authors declare no conflicts of interest.