Theoretical Investigation Of The Spontaneous Emission Of An Atom Embedded In PBG Reservoirs

In this thesis for doctorate that consists of two parts, we study the effects of the fine structure of the lower levels on the spontaneous emission spectrum of a four-level atom embedded in PBG reservoirs, and the spontaneous emission cancellation from a driven four-level atom embedded in PBG reservoirs.I: Effects of the fine structure of the lower levels on the spontaneous emission spectrum of a four-level atom embedded in PBG reservoirsIn this part, we investigate in detail the effects of the fine structure of the lower levels on the spontaneous emission spectrum of a four-level atom embedded in three kinds of Reservoirs. New features of additional transparencies and additional spontaneous emission peaks, resulting from the fine structure of the lower levels of an atom, are predicted in the case of isotropic PBG modes.(1)The spontaneous emission peak corresponds to the resonant transition from the upper level |3〉to the lower level |1〉Consider a four-level atom with upper level |3〉and lower levels |2〉, |0〉and |1〉, as shown in Fig.1. We assume that the transition from the upper levl |3〉to the lower level |1〉is coupled by the free vacuum mode (ω_λ), and those to the lower levels |2〉and |0〉are coupled by the double-band isotropic PBG mode, anisotropic PBG mode and free vacuum mode. In order to investigate the effects of the splitting of lower levels on the spontaneous emission spectra in PBG and free vacuum reservoirs, we plot the spontaneous emission spectra as functions of detuningδ_λin the three cases. From Fig.3, we see that the spontaneous emission peak corresponds to the resonant transition from the upper level |3〉to the lower level |1〉. This is to be expected and understandable, as the splitting will change the transitions coupled to the anisotropic PBG and free vacuum reservoirs. However, for the case of isotropic PBG reservoir, the effect of the splitting widthΔon the spontaneous emission spectrum is quite different. It is seen that additional and unexpected peaks appear besides the peaks corresponding to the resonant transition. In order to study what parameters have effect on this new feature, we plot in Fig.3 the spontaneous emission spectra for different splitting widthΔin the cases of isotropic PBG modes, anisotropic PBG modes and free vacuum modes.From Fig.2 we see that the splitting of the lower levels has no effect on the spontaneous emission spectrum in the case of the free vacuum modes. The splitting expands the width of spontaneous emission spectrum as the splitting width becomes larger in the case of anisotropic PBG modes, and has a complicated effect on spontaneous emission in the case of isotropic PBG modes. In order to investigate the effect of splitting width of lower levels on the spontaneous emission in the cases of isotropic and anisotropic PBG modes we plot the spontaneous emission spectra for different widths of forbidden gap as shown in Fig.3, and the different detuning of the upper level from the lower edge of the forbidden gap, as shown in Fig.4. From Fig.2,3,4, it can be seen that the detuning for the additional peaks in the case of isotropic PBG modes are covered in two ranges: from to and from to , and are unconcerned with other parameters. As a result, the additional transparencies and spontaneous emission peaks appear in the case of isotropic PBG modes. The origin for this feature can be traced back to forms of Laplace transform of the delayed Green function. The forms of Laplace transform of the delayed Green function involved in the spontaneous emission spectrum include the following items: .The above items show clearly that the splitting of the lower levels induces additional singularities of the Laplace transform of the delayed Green function for isotropic PBG modes. These additional singularities occur at , . This shows conclusively that the spontaneous emission peaks resulting from the splitting of the atomic ground state stem from the contribution of the splitting to the Laplace transform of the delayed Green function of the isotropic PBG modes.(2) The spontaneous emission peak corresponds to the resonant transition from the upper level |3> to the lower levels |0〉and |2〉. The spontaneous emission spectra discussed above is for the transition from the upper level |3〉to the lower level |1〉. In order to compare it with those for the transitions from the upper level |3〉to the lower levels |2〉and |0〉, we plot the spontaneous emission spectra for these transitions, as shown in Fig.5, which shows that the detuning for the additional peaks in the cases that the detuning for the additional peaks in the cases that the isotropic PBG modes are different from that discussed above. They cover the ranges fromΔ-1/2δ_c2c1 toΔ+1/2δ_c2c1 and from -Δ-1/2δ_c2c1to -Δ+1/2δ_c2c1. The origin for this feature can be traced back to the Laplace transform of the delayed Green function. But, the forms of Laplace transform of the delayed Green function involved in the spontaneous emission spectrum are given asTherefore, these additional singularities occur atΔ-1/2δ_c2c1,Δ+1/2δ_c2c1, -Δ-1/2δ_c2c1 and -Δ+1/2δ_c2c1. The spontaneous emission peaks resulting from the splitting of the atomic ground state stem from the contribution of the splitting to the Laplace transform of the delayed Green function for the isotropic PBG modes. The physical origin is that the DOS of the isotropic PBG radiation field has a sudden change at the edges of the forbidden gap. Furthermore, comparing the solid and dotted lines in Fig.6, we see that the transition from the upper level |3〉to the lower level |1〉reduces the intensity of spontaneous emission in the transitions from the upper level |3〉to the lower levels |0〉and |2〉.II: Spontaneous Emission Cancellation from a Driven Four-level Atom Embedded in Photonic CrystalsTwo models (upper-levels coupling model and lower-levels coupling model) of a four-level atom embedded in a double-band photonic crystal are adopted. The effects of spontaneous emission cancellation of such systems embedded in different reservoirs are investigated. Especially, the"trapping conditions"of such systems in PBG reservoirs have been discussed for the first time. We also investigate the quite different quantum interference effects of the lower-levels coupling model embedded in isotropic PBG reservoir.Consider a four-level atom with two upper levels |3〉and |2〉(coupled by a strong coherent field with frequencyω0 to a far above level |4〉and lower level |1〉as the first model (see Fig. 1(a)), and a four-level atom with one upper level |2> and two lower levels |0〉and |1〉(coupled by a strong coherent field with frequencyω0 to level |4〉as the second model (see Fig. 1(b)). We neglect the spontaneous decays from level |4〉to other levels, and assume that the transitions from the upper levels to the lower levels are coupled by the same reservoir, which are respectively isotropic PBG modes, anisotropic PBG modes and free-space modes.Before the discussion of the fluorescence spectra of the two atomic models embedded in PBG reservoirs, it is reasonable to study the different types of interference of the two models first. This can be made even more transparent if we show the"dressed"analogs of them, as shown in Fig.4. In the view of the dressed states, the spontaneous emission spectrum S(ωk) can be derived in the following forms: for models Fig.7(a) and Fig.7(b) respectively, where the dressed statesα,β,γreflect the coherence caused by the driving field. It is shown by inspection that for Model Fig.7(a), there are two kinds of coherence, one is the coherence caused by the driving field, the other is the quantum interference between three allowed transitions (α*(s)β(s),α*(s)γ(s)...). Thus, even a very small amount of coherent mixing of the atomic levels |2〉and |3〉in Fig.7(a) is sufficient to induce an interference effect between the spontaneous decay pathways of the excited states. Correspondingly, for Model Fig.7(b), there is only the coherence caused by the driving field, the fluorescence spectrum of Model Fig.7(b) is just the incoherent sum of three Lorentzian lines. There is also spontaneous emission cancellation in"trapping conditions"(32), but when theΩ_j is small, we can't observe this cancellation. These features can be seen from Fig.8.We also give the"trapping conditions"for two models embedded in three kinds of reservoirs. For upper-levels coupling model, the"trapping conditions"areFor lower-levels coupling model, the"trapping conditions"areThe spontaneous emission spectra are shown in fig.9. and fig.10. The spontaneous emission cancellation is achieved for both the upper-level coupling model and the lower-level coupling model. In conclusion, the spontaneous emission spectra of an atom embedded in three kinds of reservoirs have been discussed. The effect of splittings of the two lower levels on the spontaneous emission spectra of a four-level atom are investigated in detail. The spontaneous emission cancellation of a driven four-level atom embedded in PBG reservoirs has been considered for the first time.