Interacción de emisores cuánticos inmersos en cristales fotónicos

Abstract

Photonic crystals are periodic arrays of materials with different refractive indices. This modulation of refractive indices allows controlling the flow of light, leading to the development of different technological applications, such as the design of lasers, waveguides, and optical sensors. In particular, these materials have been used to improve the radiation-matter interaction meaningful in applications of quantum mechanics in the transport, processing, and storage of information and development of light sources with quantum properties. All these developments are based on two essential factors, which are the manipulation of the dispersion relation of light, that is, the allowed and disallowed wave frequencies inside this type of structures, and the possibility of confining electromagnetic fields in tiny regions of the order of the wavelength, and of controlling the degrees of freedom of these fields. Recent advances in radiation-matter interaction in nanophotonic systems, such as photonic crystals, have led to the discovery of unconventional phenomena that may become an engine for new protocols in quantum information processing. It has opened the door to exploring new physics in these systems, making it a current topic of great scientific interest. This thesis studies the properties of electromagnetic fields inside photonic crystals. From the development of different numerical, semianalytical, and analytical methods, Maxwell’s equations in these materials are solved, with which the band structure (allowed and not allowed wave frequencies) and the distribution of electromagnetic fields of different photonic crystal systems with periodicity in one and two dimensions are obtained. Based on the characteristics of these fields, the emission and interaction properties of emitters inside these photonic crystal structures are studied. In 1D crystals, the guided mode expansion (GME) method is adapted for calculating photonic bands in micropillars; using this method, it is possible to consider the effects that the finite dimensions of the structure have on the allowed and disallowed frequencies. A transfer matrix formalism is also used to implement an analytical method that evaluates the single-point Green’s function; this allows calculating the local density of states (LDOS) for a structure of periodic multilayers (1D photonic crystal) finite with a localized defect. Through the LDOS, we identified the defective mode of the structure and calculated the decay rates of emitters within the structure. In the case of two-dimensional photonic crystal slabs, two studies were performed. First, a semianalytical method was developed that combines the k.p approximation and the GME method to obtain an analytical expression of the photonic crystal modes, which is used to evaluate the Green’s function at frequencies close to a Dirac cone-like dispersion relation. Employing the two-point Green’s function, the properties of the interaction between dipolar emitters mediated by photons are studied; it was found that the interactions in these frequency regions are of long-range (decay with the distance between emitters as 1/r^γ) and also identified a trade-off mechanism between the range and magnitude of the interaction according to the positions of the emitters. In turn, it was found that the polarization of the dipole moment of the emitters plays an essential role in the interaction nature, being coherent (conservative) if the dipoles have linear polarization and being incoherent (dissipative) in almost the whole unit cell if the emitters have circular polarization. Second, a region of frequencies within the frequency bands of a photonic crystal slab that allows directional emission was considered; this region of frequencies is associated with van Hove singularities. Through the GME method, a description of the band and mode structure that explains the directionality by means of what is known as self-collimation was carried out. The effects of the position and polarization of the emitter in the selection of the directionality were studied, finding that utilizing these two parameters makes it possible to control the emission directions and the polarization of the emitted fields.

Photonic crystals are periodic arrays of materials with different refractive indices. This modulation of refractive indices allows controlling the flow of light, leading to the development of different technological applications, such as the design of lasers, waveguides, and optical sensors. In particular, these materials have been used to improve the radiation-matter interaction meaningful in applications of quantum mechanics in the transport, processing, and storage of information and development of light sources with quantum properties. All these developments are based on two essential factors, which are the manipulation of the dispersion relation of light, that is, the allowed and disallowed wave frequencies inside this type of structures, and the possibility of confining electromagnetic fields in tiny regions of the order of the wavelength, and of controlling the degrees of freedom of these fields. Recent advances in radiation-matter interaction in nanophotonic systems, such as photonic crystals, have led to the discovery of unconventional phenomena that may become an engine for new protocols in quantum information processing. It has opened the door to exploring new physics in these systems, making it a current topic of great scientific interest. This thesis studies the properties of electromagnetic fields inside photonic crystals. From the development of different numerical, semianalytical, and analytical methods, Maxwell’s equations in these materials are solved, with which the band structure (allowed and not allowed wave frequencies) and the distribution of electromagnetic fields of different photonic crystal systems with periodicity in one and two dimensions are obtained. Based on the characteristics of these fields, the emission and interaction properties of emitters inside these photonic crystal structures are studied. In 1D crystals, the guided mode expansion (GME) method is adapted for calculating photonic bands in micropillars; using this method, it is possible to consider the effects that the finite dimensions of the structure have on the allowed and disallowed frequencies. A transfer matrix formalism is also used to implement an analytical method that evaluates the single-point Green’s function; this allows calculating the local density of states (LDOS) for a structure of periodic multilayers (1D photonic crystal) finite with a localized defect. Through the LDOS, we identified the defective mode of the structure and calculated the decay rates of emitters within the structure. In the case of two-dimensional photonic crystal slabs, two studies were performed. First, a semianalytical method was developed that combines the k.p approximation and the GME method to obtain an analytical expression of the photonic crystal modes, which is used to evaluate the Green’s function at frequencies close to a Dirac cone-like dispersion relation. Employing the two-point Green’s function, the properties of the interaction between dipolar emitters mediated by photons are studied; it was found that the interactions in these frequency regions are of long-range (decay with the distance between emitters as 1/r^γ) and also identified a trade-off mechanism between the range and magnitude of the interaction according to the positions of the emitters. In turn, it was found that the polarization of the dipole moment of the emitters plays an essential role in the interaction nature, being coherent (conservative) if the dipoles have linear polarization and being incoherent (dissipative) in almost the whole unit cell if the emitters have circular polarization. Second, a region of frequencies within the frequency bands of a photonic crystal slab that allows directional emission was considered; this region of frequencies is associated with van Hove singularities. Through the GME method, a description of the band and mode structure that explains the directionality by means of what is known as self-collimation was carried out. The effects of the position and polarization of the emitter in the selection of the directionality were studied, finding that utilizing these two parameters makes it possible to control the emission directions and the polarization of the emitted fields.