The Effect Of The Spontaneously Generated Coherence On The Spectrum Of Resonance Fluorescence In A Three-level System

Coherent effects in atomic systems has been a hot topic in scientific research, Among them, SGC (Spontaneous Generated Coherence) has been a focus of attention. Compared with other coherences caused by the different coherent field, it is generated by a vacuum field coupling with the atomic energy levels. The spontaneous emission leading to SGC process is an incoherent process。The effect on the atomic system is obvious. It can significantly change the spectral characteristics of the atom, and would cause many unusual effects.SGC relates to the interference between two decay channels, and it demands the existence of two stringent conditions: First, the energy spacing between two upper adjacent levels (or lower levels) is small enough. Second, the direction of the two transition dipole moments should be non-orthogonal. In a real atom, it is difficult to find the system to satisfy these two conditions, and this makes the most research of SGC only remain in the theoretical stage. A number of SGC effects are only theoretical predictions and lack of experimental verification.This dissertation discusses the effect of the spontaneously generated coherence on the spectrum of resonance fluorescence of a three-levelΛsystem, and finds a practical level system in the atomic hyperfine levels of ⁸⁵ Rb. Using of the strong coherent field to couple of the different energy levels, we try to go on experimentally study on the coherent effect of spontaneous emission In the dressed states. The main contents can be divided into two parts.(1) The spectrum of resonance fluorescence with SGC in a three-levelΛsystem The three-level atom system we use is shown in Figure 1(a). With a strong coherent field between the energy levels｜e＞and｜g＞, the level｜e＞splits into two sub-level dressed states respectively:｜+＞=(｜e＞+｜g＞)/2,｜－＞=(｜e＞－｜g＞)/2,as Figure 1(b) shown. When the two dressed states radiate to the same ground state level spontaneously, spontaneous generated coherence (SGC) will arise. The formation of the dressed state energy levels and ground state level constitute another three-level V-type system which is equivalent to the system as shown in Figure 2. We can also note the similarities between the two systems in Figure 1 and Figure 2 from the other hand. In order to facilitate the theoretical analysis, we assumeγ 2=γ₃ in Figure 2.Using the same Rabi frequency, the laser field is coupled the transitions｜1＞→｜2＞and｜1＞→｜3＞at the same time. We can replace the levels｜2＞and｜3＞with｜s＞=(｜2＞+｜3＞)/2 and｜a＞=(｜2＞－｜3＞)/2 respectively. Then the system in Figure 2 can be written as the following form of the Hamiltonian. But our proposed plan in Figure 1(a) has the form of the Hamiltonian as follows:These two forms are very similar, so this one can also explain the two systems are equivalent. We have the model calculation for the energy structure in Figure 2 as the following. The system is obtained by the quantum optics theory and the density of the Hamiltonian matrix element equation of motion. Then according to the quantum correlation function, quantum theory and the Laplace transform, we can calculate the spontaneous emission spectrum when the two upper levels radiate to the ground state level. Fluorescence emission spectrum contains both coherent and incoherent parts. The coherent part results from the elastic Rayleigh scattering. The incoherent part results from the polarization of the dipole moment. So we only need to consider the incoherent resonance fluorescence spectrum.According to the theory,Zhou and Swain gave the non-coherent resonance fluorescence spectrum under diffent values ofω₂₃ and the SGC effect of varying degrees in the three-level V atomic system as shown in Figure 3. They gave the parameters as following:γ₃ =γ₂=γ, the detuningΔ=0, the Rabi frequencyΩ₂ =Ω₃=5γ. When the interval between the two upper levels is taken asω₂₃ =γ, the line is a trimodal structure, as shown in figure 3 (a), (b). Whenω₂₃ = 5γ. as can be seen from the figure 3 (c), (d), spectral lines is formed by 7 peaks. But the most obvious spectral feature is that, when considering the quantum interference effect between the two transitions, there is a clear ultasharp in the middle of fluorescence spectrum and the inner bands are higher than the outer ones, as shown in figure 3 (b), (d). When fixing the Rabi frequency, the energy interval smaller ,the line narrower. In exceptional circumstances, WhenΔ=ω₂₃/2, fluorescence quenching condition is met,γ₂ =γ₃=γ,Ω₂ =Ω₃=5γ,ω₂₃ = 5γ,Zhou and Swain gave the non-coherent resonance fluorescence spectrum as shown in Figure 4. We can see that as the increasing of the effect of SGC, the fluorescence spectrum has an obvious change. When dipole moments of the atomic are nearly parallel, as shown in (b), there is an ultasharp in the middle of fluorescence spectrum whenγ₂₃ =0.999. When the dipole moments are completely parallel, the phenomenon of fluorescence quenching happens whenγ₂₃ =γ, as shown in (c). Someone has given experimental proof of fluorescence quenching phenomenon when fluorescence quenching conditions are met and the dipole moments are parallel. We understand the fluorescence spectrum under the effect of different SGC in the three-level V model atomic system. Next we will simulate the fluorescence spectrum of the three-levelΛsystem in Figure 1 (a) when there is SGC effect. First, we simulate the resonance fluorescence spectrum of two-level atom. Order the Rabi frequency ofω₁ is 14MHz, and the Rabi frequency ofω₂ is 0, and we get the Mollow-like triplet, as shown in figure 5.We assumeγ_eg=0 in the theoretical model in figure 1(a). The spontaneous emission occurs only between e and 1 , so we orderγ_e1 = 3MHz,γ_eg=0 . At this point there are two coherent fields. We fix the Rabi frequency ofω₁ is 14MHz, and change the Rabi frequency ofω₂ into 6MHz,14MHz,19MHz respectively. We get the fluorescence spectrum shown in Figure 6 (a), (b), (c) as following. We observe the diagram (a), and we can see that compared with the two-level resonance fluorescence, seven peaks appear in the fluorescence spectrum, and lines narrower significantly, and the inner bands are higher than the outer ones.These characteristics are very similar with the features caused by SGC effect in figure 3(d). Therefore, it shows that there is SGC effect between the channels of the spontaneous emission from the dressed states to the ground state becaused of coupling ofω₂ .We can also observe that, with the increase ofΩ₂, the spectrum shows a series of features. The height of the outer bands reduces. The two peaks of inner bands separate gradually. The height of the inner band near the middle increases gradually.Next, we consider the situation whenγ_eg≠0 .For simple, we assumeγ_e1 =γ_eg = 1MHz.and the other parameters take the same values. We get the fluorescence spectrum shown in Figure 7 (a), (b), (c) as following. We can see that seven peaks appear in the fluorescence spectrum, and lines narrower significantly, and the inner bands are higher than the outer ones. With the increase ofΩ₂, the spectrum shows the same features as figure 6. The height of the outer bands reduces. The two peaks of inner bands separate gradually. The height of the inner band near the middle increases gradually.However, in the actual atomic levels which will be mentioned in Chapter 4, the actual relationship ofγ_e1 andγ_eg is eγ_eg 1≈3.5γ. Take into account the non-radiative relaxation rateγ_1g between |1> and | g>。Orderγ_e1 = 3.5MHz,γ_eg = 1MHz,γ_1g = 0.1MHz.We simulate the fluorescence spectrum of the three-levelΛsystem with the presence of SGC effect shown in Figure 8(a),(b),(c). We can see in the figures that in the actual levels withγ_e1≈3.5γ_eg, spectrum is slightly wider than that in the ideal situation. However, the general characteristics of spectral lines is consistent., and the line is a seven-peak structure. The spectrum changes due to the introduction of coherent. These features are very similar with the three-level V model. So it proves that the three-levelΛsystem which we studied has the effect of SGC. We will give the validation in the next chapter.In the dressed state representation to explain the system shown by fluorescence spectrum may give some of the features more clearly about the physical processes. We solve the intrinsic value of the three-level V system we consider:We examine the three dressed states formed by the sub-levels. Their relative position is determined by E_i= hλ_i(i=a,b,c). Dressed state energy levels and the corresponding fluorescence spectrum are shown in Figure 9. We can see that the incoherent resonance fluorescence spectrum consists of five peaks: the central resonance, the inner sidebands placed at frequencies±Ω_R/2 and the outer sidebands located at frequencies±Ω_R. In the dressed-state representation, the underlying physical processes are evident. Various components are associated with different transitions in the dressed states. The central peak comes from transitions between the same level of two neighboring manifolds of the dressed states and consists of a superposition of two Lorentzians, which are related to the decay rates of the populationρ_bb in the dressed state｜b＞and the population differenceρ_cc-ρ_aa of the dressed states｜a＞and｜c＞,respectively. However, one (located at－Ω_R/2) of the inner sidebands is the result of the transitions｜a＞→｜b＞and from｜b＞→｜c＞. Both transitions couple each other, so the spectral line is a superposition of two Lorentzians, which are associated with the decays of the dressed-state coherencesρ_ab±ρ_bc. The other inner sideband is associated with the transitions｜c＞→｜b＞and ｜b＞→｜a＞. This peak is also composed of two Lorentzians, but they are the results of the decays of the dressed-state coherencesρ_cb±ρ_ba. Transitions between the dressed states｜a＞and｜c＞contribute the outer sidebands.The premise of the above process is to meet the conditions for fluorescence quenching. When conditions are not satisfied with this premise, that is, under normal circumstances, we are seeking the matrix eigenvalue and eigen states through matrix determinant. Then the interval between the dressed state energy levels are not equal. Therefore, the transition between the dressed states will form a structure of seven peaks. We won't explain the details here.When it meets the conditions for fluorescence quenching and the largest coherent item, then all the atoms are trapped in the dressed state level｜b＞, and no photon radiation, so the fluorescence quenching happens. We conclude that the spectral narrowing is due to a slowly decay of the dressed-state population bbρ, which originates certainly from quantum interference between two transition pathways. When 12γis slightly less thanγ, although the number of particles on the energy level are still many, that is, the particles have a longer lifespan, there is still a small number of particles from the energy level down to relax. This slow relaxation leads to the ultrasharp much smaller thanγmagnitude in the center of the spectra. The particle relaxation rates of the levels｜a＞and｜c＞are down quickly, therefore, it forms theγmagnitude of the wide linear form. However, there are small number of particles on the level｜a＞,｜c＞. Therefore, the height of the peak formed will be lower than that when there is no SGC effect.(2) The experimental verification of SGC effect in a three-levelΛsystemSpecifically, we examine the three-levelΛsystem consisting of three ultra-fine levels in ⁸⁵Rb D1 line. As shown in Figure 10, in the coherent field under dressed state, we can think the association between the dressed state level |+> and |-> as the SGC effect. We carry out experiments on the observations.The experimental diagram is shown in Figure 11 (atom beam of light partly by top view). Because the SGC effect exist between the two almost degenerate energy levels, we observe the SGC effect only in a very narrow spectral range. It demands high spectral resolution. We use atomic beam device to produce collimated beam of rubidium atoms to reduce the impact of Doppler broadening in the experience. We need consider the impact of laser frequency drift. We use saturation absorption spectrum method of frequency stabilization to make the lightω₁ and coherent lightω₂ stabilized at the required resonant frequency.We adjust the intensity of the two laser beams to change the Rabi frequency with the attenuator. The Rabi frequencyΩ₁ of semiconductor laser is fixed to 14MHz. The atomic beam spot diameter interacting with the atomic beam is 2mm. The diameter of 899 laser interacting with the atomic beam is 3mm. Use attenuator to adjust its strength to make its Rabi frequencies to be 6MHz,14MHz,19MHz respectively. Then we collect the fluorescence data.We have used the above parameters for the theoretical simulation. Take into account the relaxation rate of the energy level｜e＞→｜g＞and the non-radiative relaxation rate between｜1＞and｜g＞. In the real levels, their relationship isγ_e1 ? 3.5γ_eg. We simulate the fluorescence spectrum when we plus two coherent field at the same time. The general characteristics of spectral lines is consistent. The spectral is a seven-peak structure. Due to the coherent introduced by the coherent field, there is a narrow line in the middle peak. And the inner bands are very clearly higher than the outer ones. These features are very similar with the three-level V model. For better comparison, the experimental results and theoretical results are compared together. The results are shown with solid line in Figure 12. The dotted line is the theoretical fluorescence spectrum.We can see that in addition to the narrow lines which need to be distinguished by high-resolution instrument.The results remaine consistent with the theoretical results. Because of the restrictions on the resolution of the experimental conditions, the outer and inner bands are not separated in Figure (b). But the spectral shape observed is consistent with the theoretical results. When the Rabi frequency ofω₂ increases, as shown in figure (c), we can clearly separate the intermediate region of the spectrum peak and the inside spectrum. Spectrum asymmetry is consistent with the theory.When the Rabi frequency ofω₂ increases continuatively, the spectrum structure becomes more apparent. In particular the right, we can see the inner bands are higher than outer. So we basically get the resonance fluorescence spectrum of three-levelΛmodel under SGC effect.This article uses the strong coherent coupling of different energy levels. We can observe the changes of resonance fluorescence spectrum under SGC in the dressed state levels of three-levelΛsystem, obtain the preliminary experimental results, and get the effect of SGC experimentally verified. This paper plays a significant role in the experimental study on SGC, and will promote the SGC effect in the development of application technology.