Theoretical Investigation Of Coherent Effects Of An Atom Embedded In PBG Reservoirs

In this thesis for doctorate that consists of two parts, we study the effects of quantum interference in spontaneous emission spectrum and probe absorption spectrum of atomic systems, which are embedded in different reservoirs.1. Spontaneous emission inhibition from a driven four-level atom in PBG reservoir.The model of a four-level atom embedded in three kinds of reservoirs is adopted, which is in turns to be the isotropic PBG reservoir, the anisotropic PBG reservoir and the free vacuum reservoir. The atom has two upper levelscoupled by the same reservoir to a common lower level and is driven by a coherent field to an auxiliary level. The effects of quantum interference in spontaneous emission spectrum of a four-level atom embedded in different reservoirs are investigated. Especially, the spontaneous emission cancellationin steady state in PBG reservoirs is discussed for the first time. It is interesting that two types of spontaneous emission cancellation, both the first type (due to the band gap of the photonic crystals) and the second type(due to the quantum interference of decay processes) are exhibited in one spectral line. Phenomenologically, we get the "trapping conditions" for the isotropic and anisotropic PBG reservoirs respectively. Recently, a wide variety of methods have been used to fabricate photonic crystals. Some of them have been proved to be rather successful. Much attention has been attracted by the application investigation of photonic crystals. Our theoreticalinvestigation will provide theoretical foundation for the experimental application of the quantum interference effects in PBG reservoirs such as the designation of optical switching, low-threshold lasing and others.Consider a four-level atom with two upper levels |3 > and |2 > (coupled by a strong coherent field with frequencyω₀ to a auxiliary level |4 >) and lower level |1 > as the model (see Figure.1). We neglect the spontaneous decays between level |4 > and other levels, and assume that the transitions from the upper levels to the lower level are coupled by the same reservoir, which are respectively isotropic PBG modes, anisotropic PBG modes and free-space modes.(1) Spectral line elimination due to the band gapAtomic coherence in a multilevel atom driven by an external coherent field can lead to many novel and unexpected effects on quantum optics as well as important technological applications. When a four-level atom with configuration of Figure. 1 is coupled to the free vacuum reservoir, the quantum interference of decay processes can result in spontaneous emission cancellation in steady state. This important result has been reported by Zhu and Hwang Lee, who have shown that the emission can be suppressed for all modes near the transition frequencies and the atom can be "trapped" in the upper states without decaying under "trapping conditions". The spontaneous emission spectrum of such a four-level atom in vacuum reservoir is shown in Figure.2(c). Whenη₃₂ = 1, quantum interference can cancel spontaneous emission (bold line), but forη₃₂ = 0 there is no cancellation (thin line).Comparing with the vacuum case, in PBG reservoir, there is an gap in the spontaneous emission spectra of Figure.2(a) and Figure.2(b). This new feature results from no allowed propagating electromagnetic modes for the range of frequencies of PBG. In our deduction, the influence of the photonic band gap on the atom-radiation interaction is embedded in the memory function G(t - t'), which, in turn, is determined by the DOS of the radiation field. When we choose proper parameters to make one of the emission peak in the middle of the gap, spectral line elimination is achieved even forη₃₂ = 0. Apparently, this cancellation is not due to the quantum interference, but due to the band gap. (2)Spectral line elimination due to the quantum interference.In order to investigate the spontaneous emission cancellation due to the quantum interference, we should choose proper parameters to make sure the emission peaks are outside the band gap. Firstly, we choose the symmetric parameters, shown in Figure.3. We can achieve the spontaneous emission cancellation in PBG reservoirs.Then we choose the asymmetric parameters,ω₃₂ = 5.0,Ω₂ = 0.5,Ω₃ = 1.0,â–³₃ = 4.0,â–³₂ = -1.0,δ_21c1 = -1.5,δ_31c2 = 1.5,δ_c2c1 = 2.0, C₃(0) = (1/2)^1/2, C₂(0) = -(1/2)^1/2. As we know,γis the effective spontaneous emission rate in vacuum reservoir;βandα², which are the resonant frequency splitting for isotropic and anisotropic PBG reservoirs respectively, have the same dimension withγ. So we subtituteγof the well-known "trapping conditions" in vacuum reservoir, withβandα² for isotropic PBG and anisotropic PBG reservoirs respectively; i.e.,β₃₁= 1.0,β₂₁ = 0.25,α₃₁² = 1.0,α₂₁² = 0.25,γ₃₁ = 1.0,γ₂₁ = 0.25. Comparing with the vacuum case, we can't achieve the spontaneous emission cancellation in PBG reservoirs. Conversely, the spontaneous emission peaks are enhanced for both the isotropic and the anisotropic PBG reservoirs. Here we want to stress that when we choose proper parameters, we can also achieve the spontaneous emission enhancement due to the quantum interference.Then we change the values of the parametersβ₃₁,β₂₁ ,α₃₁,α₂₁ and keep all the other parameters to explore the effects of the parametersβ_j1,α_j1² (j= 2,3) on the spontaneous emission cancellation. We find whenβ₂₁^3/2 = 0.25,β₃₁^3/2= 1.0;α₂₁ = 0.25 ,α₃₁ = 1.0, we can also achieve the spontaneous emission cancellation in PBG reservoirs. In fact the physical origin ofγis the atomic dipole moment unit vector d. And as shown in the expressions of ,β^3/2 andα, they have the same factor d² withγ. This meansβ^3/2 andαhave the same physical essence withγon the effect of spontaneous emission cancellation. Phenomenologically, we get the ''trapping conditions" for the isotropic and anisotropic PBG reservoirs respectively.(3) Explanation in dressed picture.It is reasonable to study the effects of quantum interference of the model first. This can be made even more transparent if we show the "dressed" analogs of the atom, as shown in Figure.6.In the view of the dressed states, the spontaneous emission spectrum S(ω_k) can be derived in the following forms:where the dressed states |α>, |β>, |γ> reflect the coherent caused by the driving field. It is shown by inspection that, 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 level |2 > and |3 > is sufficient to induce an interference effect between the spontaneous decay pathways of the excited states.2. Probe absorption spectra of a three-level atom embedded in PBG reservoirs.We discuss the absorption spectra of two different types of atom embeddedin PBG reservoirs in this part.(1) We introduce the "decay rate" terms into the density matrix equationsof an atom embedded in a photonic band gap reservoir successfully. By utilizing the master equations, the probe absorption spectra a threelevelatom in the PBG reservoir are obtained. The interaction between the atom and the PBG reservoir as well as the effects of the quantum interferenceon the absorption of the atom has also been taken into account. The methodology used here can be applied to theoretical investigation of quantum interference effects of other atomic models embedded in a PBG reservoir. Consider a three-level atom as shown in Figure.8, the transition |2 > -|1 > is coupled to the PBG reservoir, while the additional transitions |3 > -|1 > is induced by a classical field with an assigned Rabi frequencyΩ_R. The simplified treatment omits the spontaneous decay processes of |3> -|2 > and |3 > -|1 >.(a)Ω_R = 0 caseAs is well known, the probe absorption spectrum of such a two-level atom in a vacuum reservoir is a Lorentz line, as shown in Figure.8(b). Comparedwith the vacuum case, in the PBG reservoir, there is an absorption gap in the absorption spectrum. This new feature results from non-allowed propagating electromagnetic modes for the range of frequencies of PBG. In our deduction, the influence of the photonic band gap on the atom-radiation interaction is embedded in the memory function G(t- t'), which is in turn determined by the DOS of the radiation field.(b)Ω_R≠0 case In this case, the "dressed" analogue of the system is shown in Figure.9. The absorption spectra are shown in Figure. 10. In the vacuum case, the model has been investigated intensively for the particular property: absorptionreduction at the resonant point. In anisotropic PBG reservoir, when the photonic band gap is in the middle of two absorption peaks, the absorptionreduct to 0 at the resonant point. Obviously, this is due to the band gap. not due to the quantum interference. We can also modify the spectral structure by changing the relative positions of |2 > from the edges of the forbidden gap. When one of the dressed doublet is in the forbidden gap, the corresponding transition peak vanishes, as shown in Figure. 10.(2) The probe absorption spectra of a V-type atom embedded in PBG reservoirs have been investigated under conditions that quantum interferenceamong decay channels is important. The effect of the probe polarizationon the absorption amplitude and spectral structure is investigated in detail. Comparing with similar models located in vacuum reservoir studiedearlier, the study here shows that the quantum interference has some different effects on the absorption spectra in PBG reservoirs.In order to describe the absorption properties under different parameters,we divide this section into three subsections according to the choice of theθ.(a)θ=Ï€/2Since we choose the conditionθ=Ï€/2, quantum interference does not occure under this situation. Therefore, the spectrum is the sum of two independent Lorentzian lines. For the vacuum case, we neglect the frequency shifts of atomic levels resulting from interaction with the vacuum field in our calculation. Whenω₂₁ = 0, the two Lorentzian lines are both centred atâ–³_p = 0 . And it is to be noted that the peak value of the absorption A(0) is equal to 2, independent on the probe polarization direction.Comparing with the vacuum case, in PBG reservoir, there is an absorp- tion gap in the absorption spectrum (Figure 12(a)). Like the model studied before, this new feature results from no allowed propagating electromagneticmodes for the range of frequencies of PBG. Also, it is to be noted that the location of the absorption peak is correlated withαfor the PBG case. Whenα₁₀=α₂₀, the two absorption peaks are both centred at the same point. However, forα₁₀≠α₂₀, there is a distance between the two peaks corresponding to the two absorption transitions. And, it is interesting that the spectrum of the atom located in PBG reservoir forω₂₁ = 0 (the thin line in Figure 12(a)) is much similar to the spectral line of the vacuum case forω₂₁≠0 (the dash line in Figure 12(b)).(b)θ= 0This transparency reported here is certainly due to the effect of quantuminterference. It: is to be noted most importantly that there is a super narrow hole bored into the broad spectrum. The super narrow hole and transparency originates from quantum interference. Whenγ₁₀=γ₂₀ orα₁₀=α₂₀, the linewidth of the hole can be greatly narrowed asω₂₁→0, as shown in Figure 13. However, for the vacuum case, whenω₂₁ becomes zero, the super narrow hole disappears, it is surprising to see that the absorptionspectrum is a single Lorentzian line (Figure 14). This conclusion is also attributed to quantum interference: the two absorption transitions |1 > -|0 > and |2 >-|0 > are totally correlated and thus indistinguishable in this situation. As a result, we can regard the transitions |1 > -|0 > and 12 >-|0 > as a single transition. This is the same situation for the case of PBG, whenα₁₀=α₂₀. However, forα₁₀≠α₂₀, there is a distance between the two peaks corresponding to the two absorption transitions, and the absorptiontransitions |1 > -|0 > and |2 >-|0 > in PBG reservoir can't, be regarded as a single transition. Then the absorption spectrum is not a single Lorentzian line any more (shown in Figure 14(a)). Most interestingly, it is to be noted that there is a transparency point in the broad spectrum due to the effects of quantum interference. That is to say the two transitions are still distinguishable due to the frequency shifts of atomic levels resulting from interaction with the PBG reservoir.For simplicity , we investigate here forω₂₁≠0,α₁₀ =α₂₀ case only. The discussion aboutω₂₁ = 0 andα₁₀≠α₂₀ case is almost the same with the above section. It is clear that the quantum interference results in a super narrow hole and absorption reduction, for both of the anisotropic PBG reservoir and the vacuum reservoir. Whenθ- 0, i.e. the quantum interference is maximal, the transparency phenomenon takes place. Asθincreases, the absorption reduction becomes smaller. Finally, whenθ=Ï€/2, the narrow hole disappears and the absorption spectrum becomes the sum of two independent Lorentzian lines (shown in Figure 15).