| The interaction between photons and matter is one of the most important academic areas. As a result of the valuable applications in the high precision measure and the quantum information, the atomic spontaneous emission becomes the central issue. However, how to control the atomic spontaneous emission is the most challenge which the scientists face. With the development of the Quantum Radiation Theory, people realize that the fundamental quantum process is related with environment, which is called reservoir. Then there forms a total physical system which is consisting of "an atom" and "environment". After 1980s, a new photonic material---photonic crystal (PC) appeared. Photonic crystal is an artificial periodic dielectric material and this non-Markovian reservoir has a special density of states different from the vacuum. So there will many novel phenomena and laws in atomic spontaneous emission properties when the atom is embedded in the photonic crystal. In this thesis, we study the radiation properties of both a tripod atom driven by an external field and a there-level atom driven by a microwave field embedded in the anisotropic photonic crystal. Then the quantum entanglement between the two-level atom and its radiation field in the double-band anisotropic photonic crystal is found. Our work helps people to deeply understand the theory of the atomic dynamics in photonic crystal and achieve the purpose of controlling the spontaneous emission. What's more, as a result of the localized effect of the photonic forbidden gap, long time quantum entanglement is meaningful for the area of the quantum information.Fist, we consider a tripod atom, in which one of the three transitions is coupled by an external field and the other two transitions are spontaneously coupled to the anisotropic photonic crystal. With the Schrodinger equation in free picture in quantum mechanism and the Laplace transformation method, the atomic amplitude is calculated, and then the functions of the spontaneously emitted field, population of the two driven levels and the spontaneous emission spectra are obtained. According to the relative position of the upper level of the atom from the band edge and the space between two lower levels, we find that the spontaneously emitted field which has three kinds of type:localized field, diffusion field and travelling field, is of the regular distribution. Meanwhile, the changes of the upper level and the properties of the spontaneous emission spectra are consistent with the distribution of the spontaneously emitted fields. The dressed states form with the affection of external field and the localized effect of the forbidden band gap. The quantum interference between the two dressed states to the same lower level can lead the population's oscillation and the cancellation of the spontaneous emission.Second, we consider a three-level atom embedded in the anisotropic photonic crystal. Two upper levels of the atom are coupled by a microwave field with the intensity and phase information. With the Schrodinger equation in the interaction picture in quantum mechanism and the Laplace transformation method, the atomic excited level's amplitude are calculated, then the distribution of the spontaneously emitted fields can be obtained. After the numerical calculation, we find the distribution law of the spontaneously emitted fields is summarized according the relative position of the resonant frequency to the band edge and the intensity of the coherent field, and it has no relationship with the microwave field's phase. We discuss the effects of the excited level's population and the properties of spontaneous emission spectra influenced by the anisotropic dispersion, the microwave field's phase and intensity, and the initial population. It is found that when the initial population, the microwave field's phase and the detuning between the microwave field and the coupled levels fit a certain condition, the atomic spontaneous emission spectra appear the fluorescence quenching or spontaneous enhancement which is caused by the quantum destructive and instructive interference, and then the steady-population or decay population of the atomic excited level will appear. Meanwhile, population exchange between the two upper levels and spontaneous emission spectra are both present estimated periodic oscillation per 2Ï€. They are also consistent with the distribution of the spontaneously emitted fields.Commonly, single band gap in the photonic crystal is chosen as a good model when the effect of the lower band can be omitted. However, if the band gap is narrow, we must consider both upper and lower bands. In the last part of this thesis, we study the entanglement of a two-level atom embedded in an anisotropic double-band photonic crystal. With the Schrodinger equation in the interaction picture in quantum mechanism and the Laplace transformation method, the excited level's amplitude is obtained, and the method of the quantum entropy is used to measure the degree of the atom and its spontaneously emitted photon. Different from the case in single-band isotropic PC or in vacuum, during the process of numerical calculation, there are two no root areas, which means there are only diffusion fields. When the steady-state population happens, the atom and its spontaneously emitted photon will entangle after a long time. When the atom is near the band edge, there will be only diffusion field and the population will be trapped in the excited level, then the entanglement will achieve its maximum. But as the result of slower spontaneously rate, the time of the entanglement is a bit longer. The entanglement in the double-band PC has symmetrical properties. The degree of entanglement achieves the minimum value 0.7 in the center of forbidden gap while it is maximal near either band edge then sharply reduces to zero on the boundary of regionâ…¡. With the method of modulating the width of the photonic crystal's forbidden band gap, we can control the entanglement between the atom and its spontaneously emitted photon.
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