Abstract
We present the surface plasmon-mediated energy transfer between two optically active ions in vacuum near a metallic surface using methods of molecular quantum electrodynamics. We have studied the electric dipole-electric dipole energy transfer process only, this being the most dominant mechanism in interionic interaction between two ions in a medium when their wavefunctions do not significantly overlap. The matrix elements for energy transfer, hence energy transfer rates, are calculated using two classes of Feynman diagrams. The intermediate states for one class of diagrams do not satisfy the energy conservation principle, hence they are purely virtual states. Of particular interest are the dependencies of the energy transfer process on (1) the positions of the ions with respect to one another projected on the interface and (2) the distance of each ion from the metal surface. The overall energy transfer process has been found to have both the short range and long-range components, the former being driven by virtual plasmons and the latter by the real plasmons in a non-lossy medium.
The dependence of the energy transfer process on the relative separation of the ions projected on the interface is a function of the relative separation, ρ, and the surface plasmon wavevector, p. The transfer rate is proportional to a function F(pρ). In order to observe the behavior of this function, we chose a case of the photon of energy corresponding to a wavelength of 500 nm. For a silver interface, the dielectric constant corresponding to this wavelength is -9.
For any given wavevector, p, the plot of F(pρ) in Fig. 1 shows that the maximum energy transfer rate occurs when ρ=0; that is, when ion A and ion B have the same x and y coordinates, only differing in their z coordinates. Thus, the maximum transfer rate occurs when one ion is "on top" of the other as observed from nearest point on the plane of the interface.
As ρ increases, the energy transfer rate initially decreases rapidly. At longer distances, the transfer rate continues to decrease, but much more slowly. Oscillations in F(pρ) are present at large and small values of ρ. The "fast" and "slow" rates of decrease in F(pρ) indicate the presence of both short-range and long-range transfer mechanisms. To estimate the distance of the short-range mechanism, let us assume a surface plasmon of wavelength 500 nm, so p~2π/500 nm-1. Using the Fig. 1, we estimate that the "fast" component of the transfer rate extends to p~ 2.5, corresponding to ρ~ 200 nm, which is less than one wavelength. When pρ = 50, however, we find that ρ~ 4000 nm, which is several wavelengths long. Our interpretation is that the short-range mechanism results from energy transfer mediated by primarily by virtual surface plasmons, and the long-range mechanism corresponds formally to energy transfer mediated by primarily by real surface plasmons. Moving from small ρ to large ρ, the mediating surface plasmons change from virtual to real in a continuous fashion. We note here that we have considered a loss-less medium. In a lossy medium, the long-range interactions will be inhibited.
Figure 1. A plot of F(pρ) (black line) normalized to unity at ρ=0. For comparison, also shown is the plot of [J0 (pρ)]2. (red line).
Figure 1