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.
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.
Dipole-dipole interaction between two two-level `atoms' in photonic crystal nanocavity is investigated based on finite-difference time domain algorithm. This method includes both real and virtual photon effects and can be applied for dipoles with different transition frequencies in both weak and strong coupling regimes. Numerical validations have been made for dipoles in vacuum and in an ideal planar microcavity. For dipoles located in photonic crystal nanocavity, it is found that the cooperative decay parameters and the dipole-dipole interaction potential strongly depend on the following four factors: the atomic position, the atomic transition frequency, the resonance frequency, and the cavity quality factor. Properly arranging the positions of the two atoms, we can acquire equal value of the cooperative decay parameters and the local coupling strength. Large cooperative decay parameters can be achieved when transition frequency is equal to the resonance frequency. For transition frequency varying in a domain of the cavity linewidth around the resonance frequency, dipole-dipole interaction potential changes continuously from attractive to repulsive case. Larger value and sharper change of cooperative parameters and dipole-dipole interaction can be obtained for higher quality factor. Our results provide some manipulative approaches for dipole-dipole interaction with potential application in various fields such as quantum computation and quantum information processing based on solid state nanocavity and quantum dot system.
Quantum properties like coherence and entanglement can lead to enhanced performance characteristics in a wide range of applications including quantum computation, quantum memory storage, optical sensing, and energy harvesting. Entanglement is very sensitive to static and dynamical disorder. Similarly, the generation of highly-entangled states requires strong coupling or strong driving fields. Satisfying all of these requirements is generally quite difficult. In the first part of this thesis, we present an approach to overcome these limitations through the use of exotic light-matter states in hyperbolic media which provide a new approach to control quantum correlations and interatomic interactions. We reveal a class of excited-state, long-range interactions, referred to as Super-Coulombic interactions that are singular along a material-dependent resonance angle. In practical systems, the Super-Coulombic interaction achieves dipole-dipole coupling that is orders of magnitude larger than conventional approaches, while also occurring across a large frequency bandwidth making it robust to static energy-level disorder. This unique hyperbolic response is not only naturally occurring, found in materials like h-BN, BiTe2, BiSe2, and mono-layered black phosphorus, but can also be designed with artificial nanostructured materials (metamaterials) to create the desired hyperbolic dispersion across different parts of the electromagnetic spectrum. Our theoretical prediction motivated an intense search for the effect and was confirmed by an experimental demonstration at room temperature. To obtain agreement with experimental results, we present a rigorous theoretical framework that takes into account ensemble effects, finite-sized effects, and dimensional effects that arise from confined geometries ultimately modifying the Super-Coulombic spatial scaling law. In the second part of this thesis, we solve an outstanding theoretical problem dealing with the control of resonance energy transfer in nanophotonic environments in both the incoherent and coherent coupling limits. Resonance energy transfer is a fundamental process that is the subject of intense research across all sciences. For example, in chemistry for drug delivery and chemical monitoring, in engineering for photovoltatic and up-conversion devices, and in biology for exciton transport within photosynthetic complexes. First, we consider the disordered and weak coupling limit of resonance energy transfer often encountered in chemistry. We propose new design principles for enhancing and suppressing the energy transfer rate and efficiency quantitatively captured by a simple image dipole model. Our theory explains a wide range of experimental results which have been the subject of an ongoing debate for the past 15 years. Second, we present our recent result aimed at understanding the fundamental role of entanglement and quantum coherence in resonance energy transfer. To uncover the role of these effects, we develop a unified theory of energy transfer valid from the incoherent to quantum coherent coupling regimes. Ultimately, our theory reveals a fundamental bound ηmax = γa for energy transfer efficiency arising from γd+γa the spontaneous emission rates γd and γa of the donor and acceptor. This bound provides an upper limit to the efficiency of energy transfer regardless of quantum coherence or entanglement, suggesting new design principles for achieving near-unity energy transfer efficiency in coherent systems. The result has important implications for the two-chromophore model found in photosynthetic complexes and paves the way for nanophotonic analogues of efficiency-enhancing environments mimicking biological photosynthetic systems.
Within the framework of quantization of the macroscopic electromagnetic field, equations of motion and an effective Hamiltonian for treating both the resonant dipole-dipole interaction between two-level atoms and the resonant atom-field interaction are derived, which can suitably be used for studying the influence of arbitrary dispersing and absorbing material surroundings on these interactions. The theory is applied to the study of the transient behavior of two atoms that initially share a single excitation, with special emphasis on the role of the two competing processes of virtual- and real-photon exchange in the energy transfer between the atoms. In particular, it is shown that for weak atom-field interaction there is a time window, where the energy transfer follows a rate regime of the type obtained by ordinary second-order perturbation theory. Finally, it is shown that the resonant dipole-dipole interaction can change the singlet line of the emitted light to a doublet spectrum for weak atom-field interaction and the doublet spectrum to a triplet spectrum for strong atom-field interaction.
This chapter discusses light–matter interactions from a semiclassical point of view. By expanding the electromagnetic field into a Taylor series we derive the multipolar interaction potential and particle-field Hamiltonian. Then, using the Green function formalism, we calculate the fields of an oscillating dipole and, based on Poynting’s theorem, derive a general expression for the rate of energy dissipation in an arbitrary environment. This expression leads to the concept of local density of states (LDOS) and provides a direct link to spontaneous emission and atomic decay rates. The rate of energy dissipation of an oscillating dipole is also used to derive the absorption cross-section in terms of the polarizability. By accounting for radiation reaction and scattering losses, we obtain a compact expression for the dynamic polarizability. Dipole radiation and atomic decay rates can be enhanced via LDOS engineering, and the enhancement factor is referred to as the Purcell factor. We show that if the LDOS gets enhanced in a certain frequency range, it must be reduced in other frequency ranges, a feature described by the LDOS sum rule. After discussing the properties of a single dipole, we continue with analyzing the interaction between multiple dipoles. We derive the interaction potential and calculate the energy transfer rate between dipoles. For short distances we recover the famous Förster energy transfer formula. If the interaction energy becomes sufficiently large, we enter the regime of strong coupling, which gives rise to hybridized and delocalized modes, level splittings, and entanglement.
We derive a rigorous theory of the interaction between photons and spatially extended excitons confined in quantum dots in inhomogeneous photonic materials. We show that beyond the dipole approximation, the radiative decay rate is proportional to a nonlocal interaction function, which describes the interaction between light and spatially extended excitons. In this regime, light and matter degrees of freedom cannot be separated and a complex interplay between the nanostructured optical environment and the exciton envelope function emerges. We illustrate this by specific examples and derive a series of important analytical relations, which are useful for applying the formalism to practical problems. In the dipole limit, the decay rate is proportional to the projected local density of optical states, and we obtain the strong and weak confinement regimes as special cases.
Within the framework of quantization of a macroscopic electromagnetic field, a master equation describing both the resonant dipole-dipole interaction (RDDI) and the resonant atom-field interaction in the presence of dispersing and absorbing macroscopic bodies is derived, with the relevant couplings being expressed in terms of the surroundings-assisted Green’s tensor. It is shown that, under certain conditions, the RDDI can be regarded as being governed by an effective Hamiltonian. The theory, which applies to both weak and strong atom-field coupling, is used to study the resonant energy exchange between two (two-level) atoms initially sharing a single excitation. In particular, it is shown that, in the regime of weak atom-field coupling, there is a time window where the energy transfer follows a transfer-rate law of the type obtained by ordinary second-order perturbation theory. Finally, the spectrum of the light emitted during the energy transfer is studied and the line splittings are discussed.