光子带隙热库中原子自发辐射荧光的理论研究
摘要
近年来,自发辐射的量子相干又吸引了人们的注意力。自发辐射的量子相干是由于不可分辨的延迟通道导致的,自发辐射不仅与原子自身的属性有关,而且同作用于它的热库系统有关。在非马尔可夫热库中,有着与真空场不同的态密度,光子晶体就是这种热库。
到目前所知,早期关于光子晶体中原子自发辐射的理论研究主要将注意力放在布居数的变化和分布上,其中在真空场中由于量子相干,会导致粒子数的振荡,而采用耦合激光场并将原子置于光子晶体中还可获得稳态非零布居数。最近有很多讨论自发谱的工作,Knight和他的小组研究了光子晶体中Λ模型的透明和相干现象,其中一支跃迁同真空场相互作用,另外一支同光带隙边缘的模式相互作用。本论文采用新的方法,越过粒子数随时间的变化求解,利用Laplace变换直接得到PBG中四能级原子自发辐射谱,并同耦合到真空场模的情况进行了对比。本论文分五个部分,具体内容如下:
第一部分:介绍光子晶体的相关知识,光子晶体的制备方法、分类及研究的历史和现状。
第二部分:介绍原子自发辐射理论及描述原子自发辐射相关理论。包括描述量子系统的三种基本图像,处理光与物质相互作用问题的半经典理论和全量子理论,分析原子相干现象常用的缀饰态理论及计算荧光谱的拉普拉斯变换方法。
第三部分:介绍马尔可夫热库和非马尔可夫热库下荧光的处理方法及相应的研究现状和典型的现象、结论。
第四部分:研究在各向同性光子晶体带隙热库、各向异性光子晶体带隙热库和真空场热库中,四能级原子模型下能级劈裂对原子自发辐射谱的影响。下能级劈裂导致在各向同性光子晶体带隙热库中原子自发辐射谱出现一些额外的峰和额外的透明点。研究表明,该性质的起源可以追溯到在自发辐射谱计算中引入的延迟格林函数的拉普拉斯变换表达式上,下能级劈裂使得在延迟格林函数的拉普拉斯变换中引入了额外的奇点,正是这些奇点又导致了额外的透明点和额外的自发辐射峰的产生。
第五部分:研究在各向同性光子晶体带隙热库、各向异性光子晶体带隙热库和真空场热库中,由相干场驱动的四能级原子模型的自发辐射谱及其中的量子相干效应。我们研究了两种不同原子模型的自发辐射相消现象,探讨了两种模型中不同的量子干涉效应,分别给出三种不同热库中在两种模型中实现自发辐射相消的粒子数俘获条件。
本论文第四和第五部分的理论研究成果对光子晶体带隙热库中自发辐射的研究具有重要意义,有助于进一步发展光子晶体带隙热库中荧光的应用研究。
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 reservoirs
In 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_(c2c1)to -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 as
Therefore, 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 Crystals
Two 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”are
For lower-levels coupling model, the“trapping conditions”are
The 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.
到目前所知,早期关于光子晶体中原子自发辐射的理论研究主要将注意力放在布居数的变化和分布上,其中在真空场中由于量子相干,会导致粒子数的振荡,而采用耦合激光场并将原子置于光子晶体中还可获得稳态非零布居数。最近有很多讨论自发谱的工作,Knight和他的小组研究了光子晶体中Λ模型的透明和相干现象,其中一支跃迁同真空场相互作用,另外一支同光带隙边缘的模式相互作用。本论文采用新的方法,越过粒子数随时间的变化求解,利用Laplace变换直接得到PBG中四能级原子自发辐射谱,并同耦合到真空场模的情况进行了对比。本论文分五个部分,具体内容如下:
第一部分:介绍光子晶体的相关知识,光子晶体的制备方法、分类及研究的历史和现状。
第二部分:介绍原子自发辐射理论及描述原子自发辐射相关理论。包括描述量子系统的三种基本图像,处理光与物质相互作用问题的半经典理论和全量子理论,分析原子相干现象常用的缀饰态理论及计算荧光谱的拉普拉斯变换方法。
第三部分:介绍马尔可夫热库和非马尔可夫热库下荧光的处理方法及相应的研究现状和典型的现象、结论。
第四部分:研究在各向同性光子晶体带隙热库、各向异性光子晶体带隙热库和真空场热库中,四能级原子模型下能级劈裂对原子自发辐射谱的影响。下能级劈裂导致在各向同性光子晶体带隙热库中原子自发辐射谱出现一些额外的峰和额外的透明点。研究表明,该性质的起源可以追溯到在自发辐射谱计算中引入的延迟格林函数的拉普拉斯变换表达式上,下能级劈裂使得在延迟格林函数的拉普拉斯变换中引入了额外的奇点,正是这些奇点又导致了额外的透明点和额外的自发辐射峰的产生。
第五部分:研究在各向同性光子晶体带隙热库、各向异性光子晶体带隙热库和真空场热库中,由相干场驱动的四能级原子模型的自发辐射谱及其中的量子相干效应。我们研究了两种不同原子模型的自发辐射相消现象,探讨了两种模型中不同的量子干涉效应,分别给出三种不同热库中在两种模型中实现自发辐射相消的粒子数俘获条件。
本论文第四和第五部分的理论研究成果对光子晶体带隙热库中自发辐射的研究具有重要意义,有助于进一步发展光子晶体带隙热库中荧光的应用研究。
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 reservoirs
In 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_(c2c1)to -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 as
Therefore, 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 Crystals
Two 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”are
For lower-levels coupling model, the“trapping conditions”are
The 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.
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