Energy transfer plays an essential role in many processes in nature, such as photosynthesis in the light-harvesting complex of green plants [1,2]. Except for interpreting the energy transfer in nature, manipulating the energy transfer in artificial structures also has many potential applications. For example, white-light emitting nanofiber has been fabricated through efficient energy transfer [3], sensitive and efficient biosensing has been realized by designing novel nanostructures [4], and efficient solar energy conversion can also be achieved by plasmonic nanostructures [5].

In recent years, it has been reported that coherence may play a key role in energy transfer [6–14], and quantum coherence could enhance the efficiency of energy transfer [15–23]. However, under natural circumstances, coherence decreases quickly with the increase in distance; thus, energy transfer efficiency decreases sharply as the distance between two molecules increases, which greatly limits its application in various aspects. How to realize long-range energy transfer with high efficiency is a challenge for us. In addition, even if the long-range energy transfer with high efficiency can be realized, how to keep it stable and free from environmental perturbations is also a problem.

New developments in the field of topological photonics [24–27] make it possible to solve the above-mentioned problems. By introducing topology into the field of optics, numerous fascinating properties, such as the unidirectional backscattering-immune propagation of photonic edge modes [28–37], can be achieved. The combination of quantum mechanics and topology can bring more intriguing phenomena [38–48], including topological sources of quantum light [39–41], quantum interference of topological states of light [43], topological quantum optics interface [38], topologically robust transport of entangled photons [42], and so on. Recently, a new class of higher-order topological insulators has been theoretically proposed and experimentally demonstrated in many different systems [49–68]. Moreover, the zero-dimensional (0D) topological corner state has been observed in the two-dimensional (2D) photonic crystal (PhC) slab [55,56,67,68], which provides an ideal platform to design topological nanocavities [67,68].

Motivated by the above investigations, in this work, we provide a scheme to realize long-range energy transfer with high efficiency in the PhC platform. The designed PhC platform possesses a topologically protected edge state and 0D topological corner cavities. When the donor and the acceptor are put into two topological cavities, the energy transfer between them can occur through the topologically protected edge state. Such an energy transfer can always exist, even though the donor and the acceptor are separated by a long distance, and it is robust against disorders.

We consider a PhC platform that contains five kinds of integrable modules (I–V) in different regions, as shown in Fig. 1(a). Such a platform consists of Al2O3 (ε=7.5) cylinders with various radii embedded in air background. All modules possess a triangular lattice with the lattice constant a being 1.25 μm, and the unit cell contains six Al2O3 cylinders. In order to show the system more clearly, the color of cylinders in each region is different. The PhC system is assembled inside a waveguide with two pairs of high-reflectivity metallic plates placed in the xy and yz planes, and the height of these cylinders h is taken as 0.1a, as illustrated in Fig. 1(b).

Figure 1(c) shows the schematic lattice pattern in region I. For convenience, we note that the radius of the larger (smaller) cylinders in the triangular lattice of all regions is marked with ra (rb), and the distance from the cylinder center to the center of the unit cell is marked with R. The ra (rb) and R in five modules are different. In the region I, the ra (rb) is set as 0.14 μm (0.16 μm), and R=0.371a. The lattices in regions I/II and III/IV satisfy the inversion symmetry. In such a case, two topological corner cavities can appear at the crossing of four regions, as marked by black pentagrams in Fig. 1(a) [66]. Figure 1(d) displays their eigen-energy spectrum, and a state at 133 THz inside the gap of the topological kink states appears.

In order to design the interface state whose frequency matches the resonance frequency of the corner cavity mode, the two high-reflectivity metal plates in the xy plane [marked by M1 and M2 in Figs. 1(a) and 1(b)] are set to ensure the density of interface states of the waveguide modified (or discretized) to form Fabry–Perot (FP) waveguide modes. Also, we need to design regions III and IV skillfully, so a gradual change of parameters in different rows of the lattice is applied. Taking region III as an example, the gradually varied radii of the small (large) cylinders ra (rb) are 0.14 μm (0.16 μm), 0.1425 μm (0.1575 μm), 0.145 μm (0.155 μm), 0.1475 μm (0.1525 μm), and 0.15 μm (0.15 μm) from the first to fifth rows, and R is set as 0.243a, 0.243a, 0.262a, 0.281a, and 0.3a, respectively. The corner states we designed exist in the coexistence of sublattice symmetry breaking and lattice deformation., while the design of edge states requires only lattice deformation. If we do not do the gradual change of parameters in regions III and IV, the frequency of the corner state will fall in the gap of the interface state or the bulk state, where the coupling between the corner state and the interface state cannot occur [66]. In addition, the parameters in the sixth and seventh rows are the same as in the fifth row: ra=rb=0.15μm, R=0.3a (a/R>3). The ra=rb=0.15μm and R=0.36a (a/R<3) are taken in region V. In such a case, a topologically protected interface state with the working frequency located at 133 THz, which corresponds to the resonance frequency of the corner cavity mode, can be obtained [36,37]. This can be demonstrated by the calculated dispersion relation presented in Fig. 1(e). This gap is a result of the avoided crossing of the interface states caused by their interaction due to the broken C6 symmetry at the conducting interface. Theoretical and experimental results show that the topological interface mode is still extremely robust, and the backscattering can be ignored [37,69–71]. The distance between two topological corner cavities is L2, and the distance between the left (right) cavity and the left (right) metallic plates is L1 (L3). Here, we consider the transverse magnetic mode with the out-of-plane electric field and the in-plane magnetic field.

We put a pair of two-level molecules into the topological corner cavities. The dynamic process and energy transfer properties between two molecules are studied through a master equation approach. Initially, one of the molecules is excited, and the other in its ground state; they are called donor (D) and acceptor (A), respectively. The corresponding Lindblad master equation of two molecules [19,72–75], whose microscopic origin and derivation using the Born–Markov and rotating-wave approximations can be found in Refs. [73–75], is expressed as follows: $$\begin{gathered} ∂tρ=iℏ[ρ,H]+γD+2γ02(2σDρσD†−σD†σDρ−ρσD†σD) \\ +γA+2κ2(2σAρσA†−σA†σAρ−ρσA†σA) \\ +γ122(2σDρσA†−σD†σAρ−ρσD†σA) \\ +γ212(2σAρσD†−σA†σDρ−ρσA†σD), \end{gathered}$$(1)where ρ is the density matrix of two molecules, and σD(A)† and σD(A) are, respectively, the creation and annihilation operators for the donor (acceptor). γ0 is the natural dissipation in the donor, and κ is the separation rate in the acceptor, and in this work, we have taken γ0=1ns−1 and κ=1ps−1 in our calculations [19]. Due to the symmetry of two cavities, we assume the decay rates of two molecules are the same, namely, γD=γA=γ1. The γ12 represents the incoherent coupling term (interference term) between the two molecules, and we also have γ12=γ21. The γD and γ12 can be calculated as γD=2ω02ε0c2ℏIm[μ→D*·G↔(r→D,r→D,ω)·μ→D],(2)and γ12=2ω02ε0c2ℏIm[μ→A*·G↔(r→A,r→D,ω)·μ→D].(3)

In Eqs. (2) and (3), G↔(r→A,r→D,ω) is the classical Green tensor of the system, which is the solution to the tensor equation [∇×∇×−k02ε(r→,ω)]G↔(r→,r→′;ω)=I↔δ(r→−r→′), and r→D and r→A are position vectors of the donor and acceptor, respectively. μ→D and μ→A are dipole moments of two molecules, and in this work, we have taken |μ→D|=|μ→A|=6.4×10−23C·m. The Hamiltonian of two molecules shown in Eq. (1) can be expressed as H=∑i=D,Aℏ(ω0+δi)σi†σi+g12(σD†σA+σA†σD),(4)where g12 is the coherent coupling between two molecules, which can be calculated with g12=ω02ε0ℏc2Re[μ→A*·G↔s(r→A,r→D,ω)·μ→D].(5)

The Green tensor G↔s(r→A,r→D,ω) shown in Eq. (5) is the scattering Green tensor of the field dipole located at r→A and the source dipole located at r→D, which can be obtained with n→A·G↔s(r→A,r→D,ω)·n→D=−n→A·E→s(r→A)|r→D,(6)where n→A and n→D represent the direction of the field dipole and source dipole moment, respectively, and E→s(r→A)|r→D represents the scattering electric field at the location of field dipole r→A induced by the source dipole located at r→D.

In this work, the orientation of the source dipole moment is taken as the positive direction of the z axis, which will excite the TM modes of the designed structure, and the electric direction of the field dipole is also parallel to the z axis. In this case, the above equation turns into Gzzs(r→A,r→D,ω)=−E→s(r→A)|r→D.(7)

Note that an important Markov approximation has been employed in the derivation of Eq. (1), which requires that the coherent couplings between a single molecule and electromagnetic modes are much weaker than the couplings between molecules through the electromagnetic field. In the present work, the main conclusion is obtained when the coherent coupling between the two molecules is much larger than the coupling between the single molecule and electromagnetic field, which satisfies the Markov approximation. Thus, Eq. (1) is valid in this context. Based on Eq. (1) and the calculated parameters shown in Eqs. (2), (3), and (5), we can derive the dynamics of two molecules and investigate the coherent energy transfer between them. There are two important parameters to measure the properties of coherent energy transfer: energy transfer efficiency η and transfer time tf [19,23,76]: η=2∫0∞κPA(t)dt,(8)and tf=∫0∞tPA(t)dt/∫0∞PA(t)dt.(9)The energy transfer efficiency η represents the total probability of excitation used for charge separation, and the energy transfer time means the average waiting time before charge separation happens in the acceptor [19,23,76]. In Eqs. (8) and (9), PA(t) is the population of the acceptor and represents the probability density of the donor in the ground state, and the acceptor is in the excited state at time t, which is equal to the matrix element ρ22(t) obtained from Eq. (1) under the normal basis ({|00⟩, |01⟩, |10⟩, |11⟩}). Based on Eqs. (8) and (9), we can calculate the coherent energy transfer efficiency and time in the designed system. Note that Eqs. (8) and (9) are derived from Eq. (1), and it can be seen that both of them depend not only on the parameters of the acceptor but also on the parameters of the donor.

In this section, we present the calculated results for the coherent energy transfer efficiency and time in our designed PhC platform. One of the advantages of this platform is that the coherent strong coupling is easy to be realized. The red solid line in Fig. 2(a) displays the calculated results of the normalized coherent coupling coefficient (g12/γ11) as a function of the transition frequency. The corresponding incoherent coupling coefficient (γ12/γ11) is shown by blue dashed lines. It is seen clearly that very strong coherent coupling (g12/γ11=7.005) can be reached around the frequency f=133.0175THz; at the same time, the incoherent coupling is almost equal to zero. Under the strong coherent coupling regime, if one of the molecules is excited, the excitation will be transferred back and forth between the two molecules, namely, coherent energy transfer. The energy transfer efficiency η and time tf are calculated as a function of the frequency for the case of L2=20a, and the results are presented in Figs. 2(b) and 2(c), respectively. It is found that when coherent coupling is maximum, energy transfer efficiency can reach 0.75 [corresponding to the mark “1” in Figs. 2(a)–2(c)], which is much larger than those in other systems [15–23]. At the same time, the energy transfer time is about 200 fs, namely, fast and high-efficiency coherent energy transfer is demonstrated in the present PhC platform. In fact, for energy transfer, high transfer efficiency and short transfer time have always been our goal, which is beneficial to sensors, quantum information science, and so on.

We notice that there are two other peaks in energy transfer efficiency, where both the coherent and incoherent couplings are not zero; however, at these two frequencies, the energy transfer times are much longer than at the coherent coupling peak. The reason can be found in Fig. 2(d), where the populations of the acceptor are plotted at three typical frequencies, marked in Figs. 2(a)–2(c). In the case of maximal coherent coupling (marked by “1”), the population of the acceptor (red solid curve) reaches a relatively large value at the beginning. This is because the coherent coupling is proportional to the height of the quantum beat and the beat frequency, while incoherent coupling controls the duration of a single quantum beat. Therefore, the maximal coherent coupling and minimal incoherent coupling not only guarantee larger energy transfer efficiency, but also ensure less energy transfer time. For another efficiency peak (marked by “3”), although the integral of population over time is also large, it is clear that the energy transfer has also undergone a much longer time. The blue dashed curve represents the case in which the incoherent coupling is maximum and coherent coupling is zero (marked by “2”). Obviously, small coherent coupling leads to a very low quantum beat, which results in a small energy transfer efficiency.

In order to explore the role of coherent coupling played in energy transfer, we plot the results without coherence (g12=0) by blue dashed curves in Figs. 2(b) and 2(c). It is clearly shown that the energy transfer efficiency with coherent coupling is always larger than that without coherent coupling, and the energy transfer time is always shorter than that without coherent coupling. That is to say, coherent coupling always helps increase energy transfer efficiency while reducing energy transfer time.

The important feature of this platform is that the realized energy transfer can maintain a very long distance. The red lines in Figs. 3(a) and 3(b) represent the calculated results of coherent energy transfer efficiency and time for the case with L2=60a. For comparison, the corresponding results for the case with L2=20a are shown by black dashed lines. Obviously, the increase in distance allows only a slight decrease in efficiency and a slight increase in transfer time, except for the frequency shift presented. The frequency shift is due to the different intrinsic eigenfrequencies of different structures. This means that the present structure can be used to achieve efficient and fast energy transfer over longer distances. This is in contrast to the conventional case, where energy transfer efficiency will decrease sharply with the increase in distance between the donor and the acceptor, and efficient and rapid energy transfer occurs only at a small separation distance.

Another important feature of this platform is that the realized energy transfer is topologically protected, i.e., it is robust against various kinds of defects. In order to exhibit such a phenomenon, we consider two types of defects. One is to introduce intentionally deformed artificial atoms at the interface, and the other is to use a sharply bent interface. Here, L2=20a is taken. The red solid lines in Figs. 4(a) and 4(b) display calculated results for energy transfer efficiency and time with the first kind of defect, and the corresponding results for the second kind of defect are given in Figs. 4(d) and 4(e). For the first defect, as shown in the black dashed rectangle of Fig. 4(c), the black cylinders represent that their positions have moved 0.02a to the left, and the white cylinders represent that their radii have reduced to 0.145 μm. For the second defect, the original interface has been replaced by an interface with 60° and 120° turns, as shown in Fig. 4(f). For comparison, the corresponding results without defects are shown by black dashed lines. It can be seen that under the two defects, there is only a slight change in energy transfer efficiency and frequency shift, and the energy transfer time has a slight increase, which exhibits very good robust properties. In order to disclose the origin of these phenomena, the distributions of the electric fields at the interface with defects are plotted in Figs. 4(c) and 4(f) when chiral light sources (marked with stars) are put on the interface. It is found that there is almost no backscattering with disorders. It is the topologically protected channel that leads to the robust energy transfer. Note that we put the source at the position of the interface, not the corner cavity, just because we want to prove that the edge state is topologically protected. In fact, the energy transfer between the two corner cavities is topologically protected, which depends on the robustness of the interface states.

The above results are obtained without considering the effect of scattering loss due to roughness of the cylinders and absorption of the material. In fact, strong scattering loss and absorption do have an influence on the above phenomena [77,78]. However, when the absorption and scattering losses are small, the effect is also small. In the wavelength range we are concerned with above, the absorption of Al2O3 cylinders is quite small or even negligible, which has a limited effect on the results of energy transfer. Also, two ends of the designed PhC are metal plates, which make sure that the electromagnetic waves are difficult to scatter into the air. In addition, compared with some traditional methods to realize the interaction between two quantum dots, our method can achieve a longer distance. For example, the long distance of 1.4 μm interaction of two different InAs/GaAs quantum dots in a PhC micro-cavity has been realized [79], which is much shorter than the distance realized (tens to hundreds of micrometers) in our work. More importantly, long-range energy transfer using this scheme is robust against various kinds of defects. Therefore, the design has more obvious advantages compared with traditional ones, which is very beneficial for practical applications.

Based on the PhC platform with the 1D topologically protected edge state and 0D corner cavities, we have demonstrated that the long-range coherent energy transfer from a donor to an acceptor can be realized. Because the donor and the acceptor have been put into topological cavities, and the energy transfer between them has been fulfilled with the assistance of the topologically protected interface state, such an energy transfer is robust against various disorders and can also occur over very long distances. These results are very important for various applications, such as biological detections, sensors, and quantum information science.