Radiation-Induced Correlation between Molecules Nearby Metallic Antenna Array

We theoretically investigate optical absorption of molecules embedded nearby metallic antennas by using discrete dipole approximation method. It is found that the spectral peak of the absorption is shifted due to the radiation-induced correlation between the molecules. The most distinguishing feature of our work is to show that the shift is largely enhanced even when the individual molecules couple with localized surface plasmons near the different antennas. Specifically, we first consider the case that two sets of dimeric gold blocks with a spacing of a few nanometer are arranged, and reveal that the intensity and spectral peak of the optical absorption strongly depends on the position of the molecules. In addition, when the dimeric blocks and the molecules are periodically arranged, the peak shift is found to increase up to ~1.2 meV (300 GHz). Because the radiation-induced correlation is essential for collective photon emission, our result implies the possibility of plasmon-assisted superfluorescence in designed antenna-molecule complex systems.


I. INTRODUCTION
Optically-illuminated metallic nanostructure, such as nanoparticles and nanoblocks, generates an extremely localized electric field due to localized surface plasmon resonance (LSPR), by which we can make individual molecules effectively couple with weak light [1].
In addition, the intensity gradient of the localized field becomes steep enough to break long-wavelength approximation, and excite dipole-forbidden transitions even in a single molecule [2][3][4]. These features remind us of sophisticated antenna systems for photons, and have attracted increased attention from various scientific fields with a view to efficient reaction fields [5][6][7]. We have theoretically examined optical responses of antenna-molecule coupled systems, and found that, for certain set of parameters, one can efficiently excite only the molecule with the antenna excitation inhibited (energy transparency effect) [8,9].
Actually, similar effect is experimentally verified using gold nanoantenna system [10]. Such a transparency is explained by considering the quantum interference between the molecular polarization and the plasmon, and it can be applied also to the nonlinear response by a few photons [11].
Besides, it is theoretically proposed that the plasmon-assisted enhancement of lightmolecule interaction can be applied to cooperative photoemission such as superradiance and superfluorescence [12,13]. According to the Dicke's theory, the ensemble of emitters can radiate coherent light after creating a macroscopic dipole moment through the incident and radiation fields [14,15]. Then, in general, cooperative photoemission requires highdensity emitters. Actually, in Ref. [12,13], all the molecules are assumed to lie under the common localized field. However, if the molecules are efficiently excited by using the energy transparency effect, the radiated field from the molecule should be enlarged. Then, such a plasmon-assisted radiations can enhance inter-molecule correlation.
In the present work, we numerically analyze the optical absorption of molecules near a metallic nanostructure by using discrete dipole approximation (DDA) method [16][17][18], and investigate the correlation between the molecules. Here it is assumed that the metallic structure consists of dimeric gold nanobloks with a spacing of a few nanometer. In such a system, the intensity of the localized electric field becomes ∼ 10 5 times larger than that of the incident light [19]. In addition, the periodic array of the dimeric blocks is also produced experimentally [20,21]. It is found that the spectral peak of the optical absorption becomes largely shifted even when the molecules couple with different plasmons. We also find that radiation-induced correlation is further enhanced when the dimeric blocks and molecules are periodically arranged. The quantitative study of the inter-molecule correlation is essential to explore the cooperative photoemission in the metallic antenna-molecule coupled system, and may contribute to designing future photoemission devices.

II. MODEL AND METHOD
The system under consideration is shown in Fig. 1(a). The size of each block is 40 × 40 × 4 nm 3 , and the spacing between the blocks is 2.83 nm. For a clear demonstration, we have considered the blocks to be sufficiently thin so that the mode volume of the localized light fields becomes comparable to the molecular volume. In order to obtain the optical absorption, we solve the discretized integral form of Maxwell's equation within DDA [16][17][18]. In this calculation, we divide the whole space containing the metal blocks into small cubic cells, where microscopic quantities such as the electric field and polarization in each cell are averaged. The integral equation is given by where n represents the total number of cells, and i, j are the cell number. The incident field E 0 i is linearly polarized along the direction of the orange two-way arrows in Figs. 1(b-d), and the polarization in each cell is connected with the total electric field E i as Here, χ j and V j are the optical susceptibility and the volume of the j-th cell, respectively.
The free space Green's function G f i,j has both transverse and longitudinal electromagnetic components. The term including M i − L i denotes the self-interaction, which is introduced in order to avoid unphysical divergence and can be calculated analytically as [22,23] and Here the parameter a characterize the size of the unit cell, and k = ω/c is the wave number with ω being the frequency of the incident field.
The unit cell in the metal blocks has a Drude-type dielectric function where the background dielectric constant of the gold is set to be ε m bg = 12.0, and the environmental background dielectric constant to ε bg = 1.0, i.e., the dielectric constant of vacuum.
As for the other parameters in the gold blocks, we employ the ones in Ref. [24,25]: the plasma frequency ω p = 8.958eV, the nonradiative damping constant γ b = 72.3meV, the mean free path of electron D eff = 20nm, and the Fermi velocity ν F = 1.4 × 10 6 m/s. As for the susceptibility of the molecule, we assume it to be Lorentzian; where d = 9.7 Debye is the dipole moment, Ω = 1.57eV is the resonant energy, and γ = 1meV is the nonradiative damping constant. We have employed these parameters so that the molecular system becomes resonant to LSPR of the gold blocks, which is necessary in order to introduce the energy transparency effect [8,9].
In this work, we mainly calculate the absorption cross section of molecule according to the definition It should be noted that the total electric field is derived by numerically solving the simultaneous equations in Eqs. (1, 2), i.e., the electric fields in the metallic antenna and the molecule are self-consistently calculated. We will estimate the correlation between molecules from the spectra of the absorption cross section.

III. RESULTS
Before calculating the optical absorption of the molecules, we show the profile of the electric field intensity for the case where only the four gold blocks are set (see Figs.1(bd)). One can see that electric field is enormously enhanced nearby the gap structure of the dimeric antenna, so-called "hot spot". Therefore, when the molecules are set at the hot spot, the plasmon fields excite the molecules efficiently. Note that the electric field becomes considerably small far from the hot spots, and then we can excite plasmons with small correlations in the distant spots.
Subsequently, we investigate the correlation between the two molecules. We arrange the molecules at the position of the red and blue circles in Figs. 2(a-d). In Fig. 2( One can see the large peak shift (∼0.6meV) due to the inter-molecule interaction, only when both of the molecules are located in the hot spots ( Fig. 2 (c,d)).
line). We consider that this additional shift originates from the molecule-molecule coupling via the radiation field, because the interference between the plasmons in the distant hot spots has been already included in the case with single molecule. This is the essential difference from the previous work [12,13]. The additional shift reaches 0.6meV (150 GHz) that is comparable to the one by the antenna-molecule coupling. The large inter-molecule correlation appears only when both of the them are excited with high efficiency due to energy transparency effect, and then the plasmon-molecule interference plays the essential role also in the inter-molecule correlation. Besides, when the polarization direction of the incident field is altered, the number of hot spots is also altered. Seeing the yellow dashed line in Fig. 2 (e), the comparable shift is observed even though the absorption cross-section decreases due to the decrease of the effective electric field in one hot spot.
The inter-molecule correlation can be further enhanced by periodically arranging the dimeric antenna-molecule complex. In order to calculate the optical absorption in such a system, we apply the periodic boundary condition to DDA calculation following Ref. [26].
In order to apply the boundary condition, we have changed slightly the spacing between the nanoblocks into 3nm. Figures 3(a-d) show the arrangement of the metallic antenna and molecules. In Fig. 3(e), we plot the absorption spectra of one molecule for different arrangements. One can see that the peak shift of the absorption spectrum is no more than 0.4 meV when we add the second set of dimeric antenna and molecule. On the other hand, in case of 1D periodic array structure, the peak shift is increased up to ∼1.2 meV (300GHz), which exceeds the one due to dipole-dipole interaction between one dimeric antenna and one molecule.

IV. CONCLUSION
With the use of discrete dipole approximation method, we have numerically investigated optical absorption of molecules nearby metallic antennas. We find that large radiationinduced correlation between the molecules can appear even when they couple with different plasmons. The essence of such a large correlation is energy transparency effect due to quantum interference between the plasmon and molecular polarization. Because of the effect, one of the molecules is effectively excited and radiates a photon that couples to the other molecule. We also have evaluated the inter-molecule correlation in periodically arranged antenna-molecule complex, and found that the correlation can be comparable or greater than the single antenna-molecule correlation.
Although our calculation is based on only linear response theory, the large inter-molecule correlation makes us expect to introduce collective photoemission. Because the metallic antenna can be designed, it may lead to photoemission devices using controlled superfluorescence. For that aim, we have to calculate optical responses including nonlinear excitations in future work.

V. ACKNOWLEDGEMENTS
We thank Dr. Y. Mizumoto for helpful discussion on numerical calculation by discrete dipole approximation method. This work was partially supported by a Grant-in-Aid for JSPS Fellows No.13J09308 and a Grant-in-Aid for Scientific Research (A) No. 24244048