Abstract
Using the electromagnetic response function of an electric dipole located within a dielectric geometry, we derive the mathematical equivalence between the classical response and quantum mechanical resonant dipole–dipole interaction between two quantum objects (atoms, quantum dots, etc). Cooperative spontaneous emission likewise emerges from this equivalence. We introduce a practical numerical technique using finite difference time domain for calculating both dipole–dipole interaction and collective spontaneous emission in confined dielectric structures, where strong light–matter coupling might arise. This method is capable of obtaining resonant dipole–dipole interaction over a wide range of frequencies in a single run. Our method recaptures the results of quantum mechanical second order perturbation theory for weak light–matter coupling. In strong coupling situations such as near a photonic band edge, second order Rayleigh–Schrödinger perturbation theory leads to divergences, and instead Brillouin–Wigner perturbation theory is required. This is equivalent to the use of a variational wavefunction to describe the exciton transfer between initial and final states. We introduce a system of coupled classical oscillators, that describes resonant dipole–dipole interaction and vacuum Rabi splitting in the strong-coupling regime, and that provides an effective numerical scheme based on the finite difference time domain method. This includes the effects of quantum entanglement and the correlation of quantum fluctuations. We discuss the crossover to Forster energy transfer when quantum correlations between the dipoles are damped by strong environmental interactions.
Highlights
We have presented a simple derivation of the mathematical equivalence between weak-coupling resonant dipole–dipole interaction (RDDI) and the Green functions of Maxwell’s equation
We have further demonstrated a powerful computational algorithm that provides an accurate description of coherent RDDI in complicated strong-coupling photonic structures using finite difference time domain (FDTD) calculation
Our method supersedes simple perturbation theory in which singularities may appear at the edge of band gaps of photonic crystals or in the case of discrete cavity modes
Summary
In the absence of strong light–matter coupling, RDDI is described by second order RS quantum mechanical perturbation theory [15]. The index λ runs over all possible parameters that characterize the photonic modes. Λ is a discrete index that includes all different photonic states characterized by their mode indices. Aλ and aλ+ are the annihilation and creation operators and Eλ(r ) is the normalized electric field vector associated with the photonic mode under consideration [15], obtained by solving Maxwell’s equations under the appropriate boundary conditions. Assuming one atom is initially excited while the other is in its ground state, a straightforward treatment of Hamiltonian (1) using the RS perturbation theory yields the RDDI expression (transition matrix element for exciton exchange between the two atoms) [15].
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