| Spontaneous emission is the basic phenomena of the interactionbetween light and substance, it is an important research direction inmodern quantum optics field.It is also well known that the spontaneous emission depends notonly on the properties of the excited atomic system but also on thenature of the surrounding environment. Before the concept ofPhotonic Crystal was invented, the pioneers paid attention to alteringthe nature of the surrounding environment by effective means such asplace the atom in waveguide, microcaves etc. It is worth to point outthat they have a different density of modes compare with free-spacevacuum field because of the boundary condition, to control thebehavior of atom. While together with the new concept of Photonic,people was attracted immediately. It was not only the great modify ofdensity of modes near the transition but also an all direction band gapcan be designed and fabricated, this is thought made up theshortcoming of ordinary cave. As a result, the spontaneous emissionwas dramatic modified at the edge of Photonic Band Gap.In this thesis, spontaneous emission from three-level atom inPhotonic Crystal has been investigated. We investigate thespontaneous emission spectrum from a Λ three-level atom indouble-band PBG in detail. The two lower levels are coupled by amicrowave field. Using interact Hamiton and Schrodinger Equation,by means of Laplace transform we obtained the spontaneousspectrum equation from the probability amplitude equation varieswith time. It is show that spontaneous emission depends not only onthe properties of the external field but also on the nature of thesurrounding environment (such as vacuum field or Photonic Crystal).There are different properties of spontaneous emission invacuum field and Photonic Crystal. Dark lines and a narrowspontaneous line may be produced in isotropic PBG. The dark linesare consequences of destructive interference of the transitions andsingularities of the radiation field's DOS. In contrast, the narrowspontaneous line is a consequence of constructive interferences of thetransitions and the coupling of the transitions and the edge of PBG.The atomic spontaneous fluorescence spectrum in vacuum fieldwill be replaced by the mirror image of the old, when the phase of themicrowave field is increased by π radians. The spontaneous linesfrom will be replaced by the mirror image of the lines from. Spontaneous emission spectrum has a close relationship with| α kλ|2| βkλ|2the phase of the external field. This is caused by the interferencebetween the different paths for a radiative transition. If the frequencyof the microwave field is resonant with the transitions's, the linesbelonging to have nearly the same height, and the same istrue for the components of .| α kλ|2| βkλ|2Only when the value of coupling constant is 1, the spontaneousemission spectrum in Photonic Crystal has the relatingship of mirrorimage such as that in vacuum field. In the cases of isotropic PBG, theeffect of quantum interference on the narrow spontaneous lineincrease as the forbidden gap increase. If the frequency of themicrowave field isn't resonant with the transitions's, narrowspontaneous lines have been produced at level |2> and leve |1>, andthe height of the narrow spontaneous lines increase as the detuningincrease. The one peak in the spectrum transforms into two peaks(one is narrow spontaneous line ) with the increasing Rabi frequency;if the levels is very near the edge of PBG, the narrow spontaneousline will be very high. That is because of the transitions are coupledto the edge of PBG. In the cases of anisotropic PBG, the two peaks inthe spectrum transform into one peak with coupling constant α 321increase, and there is a very small peak producted at level |1> andlevel |2>, respectively.Compared with what has been investigated, the effects of theparameters of the system and the properties of the external field onspontaneous emission are discussed in detail in this thesis.
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