本篇論文探討了兩顆受到電磁波誘發透明效應的Lambda原子間的偶極-偶極交互作用。利用引進了等效哈密頓函數的薛丁格方程式，我們可以得出系統的穩定態解。我們探討了探測雷射光的吸收譜、穿透譜和螢光強度。在狄基模型的案例中，六個對稱態形成了一個包含三個Lambda系統的塔狀組態，而另外三個非對稱態形成了一個V型組態，且這兩個組態間無任何關聯。從綴飾態的觀點出發，藉由狀態的衰變率，我們可以理解當驅動雷射光遠離共振時的探測光吸收譜特徵。當驅動雷射光為共振時，雙原子系統的探測光吸收譜和單原子系統相比有較小的最大值和較寬的半峰值寬。我們同時也推導出此系統的暗態。在非狄基模型案例中，我們探討了原子間距對探測光穿透譜的影響。當間距等於1/8倍的波長時，此穿透譜在非共振區有四個低谷，且在共振處有一最大值，此極值即為電磁波誘發透明現象。藉由一微擾參數逐漸調大此系統的偶極-偶極間交互作用，我們可以清楚觀測出此
穿透譜逐漸由對稱的兩低谷變成四個非對稱低谷。從綴飾態的觀點出發，並且考慮偶極-偶極交互作用所造成的對稱態與非對稱態間的能階分裂，我們可以找出這四個低谷的來源。原子間的偶極-偶極間交互作用除了受到原子間距的影響，也會因不同的探測光照射方向而改變。我們探討了不同照射光方向對穿透譜的影響，並將此結果與對稱態和非對稱態間的能階分裂做出連結。我們同時也探討了在不同探測器位置上量測的探測光穿透譜及螢光強度。

In this thesis, we want to investigate the dipole-dipole interaction between two lambda-type atoms with a probe and a control laser. We use the Schrodinger equation approach with the effective Hamiltonian to get the steady-state solution. We are interested in the absorption, transmission, fluorescence intensity of the probe field. In the Dicke model case, we find that the symmetric states form a “Poker Tower” configuration, which contains three Lambda systems, decoupled from the V-type configuration formed by the other three anti-symmetric states. In the far-detuned control laser case, the feature of the absorption spectrum can be interpreted by jumping into the dressed-state picture and finding the decay rates of the participated states. In the resonant control laser case, the absorption maximum is smaller but with broader peak comparing to it in the single-atom case. We also investigate the dark states of our system in the Dicke model case. In the non-Dicke case, we find that how the spacing of the atoms affects the transmission spectrum. In r=1/8 case, the spectrum has four dips in the non-resonance regime and a maximum at resonance. By adding a perturbative term of the dipole-dipole interaction, we can clearly see that the spectrum gradually splits to four dips along with the growing dipole-dipole interaction in this case. In order to realize the origin of the four dips, we further analyze our system in the dressed-state picture. Dipole-dipole interaction is also differed by the probe light incident direction. We find the relation between the probe light incident direction and the dips locations of the transmission spectrum. The probe light transmission and fluorescence intensity measurement of different detector locations are also mentioned.

誌謝 i
摘要 ii
Abstract iii
1 Introduction 1
1.1 Motivation 1
1.2 Electromagnetically Induced Transparency 3
1.2.1 Basic Hamiltonian 3
1.2.2 Polarization and susceptibility 5
1.2.3 Absorption and transmission 5
1.2.4 The origin of EIT 6
1.2.5 Slow light 7
1.3 Dipole-dipole interaction 9
1.3.1 Heisenberg’s equation of motion 9
1.3.2 Two two-level atom 12
1.4 Thesis outline 13
2 Model and Methodology 14
2.1 Basic Hamiltonian 14
2.2 Absorption, Transmission, and Fluorescence 17
2.2.1 Absorption in the two-atom system 17
2.2.2 Transmission in the two-atom system 17
2.2.3 The fluorescence intensity of the two-atom system 19
2.3 Steady-state solution 19
2.3.1 The Schrodinger equation approach 20
2.3.2 Density matrix approach 21
3 Generalized Dicke’s picture for two L atoms 22
3.1 Steady-state solution 23
3.2 Symmetric and anti-symmetric picture 24
3.2.1 Orders of magnitude for population and coherence 25
3.2.2 Superradiance and subradiance 27
3.3 Dressed-state interpretation 28
3.3.1 Dressed-state picture of a single lambda-type atom 28
3.3.2 Dressed-state picture of two lambda-type atoms 30
3.4 Linear susceptibility of the probe light 32
3.5 Dark resonance 34
4 General cases with a two-atom array 36
4.1 Steady-state solution 36
4.2 Transmission spectrum 38
4.3 Dressed-state interpretation 41
4.3.1 Dressed-state configuration and the AC Stark shift 42
4.4 Arbitrary probing angle 44
4.5 Fluorescence 48
5 Conclusion 52
Bibliography 54

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