Theoretical And Experimental Research On Electromagnetically Induced Photonic Band-gap In Two-level Energy System

This thesis focuses on fundamental theory of atom coherence. The major content is about electromagnetically induced photonic band gap in hot Cs atom of two-level energy system, and the factors which make influence on band-gap ,such as coupling field, probe field and reflection index. In order to accomplish this topic, it need to consider Doppler broaden, Fourier transformation, density matrix and transfer matrix together.The first part: theory deductionWe imply semi-classic theory of interaction between optical field and materials, density matrix to analyze and calculate the controllable photonic band gap under the condition of Doppler Effect.It gives us simplified energy structure of Cs atom in figure 1.ΔcandΔpare detuning of coupling field and probe field between the two energy level.ΩcandΩp are Rabi frequency of coupling field and probe field. We assume that 2 was corresponded to 6 2 P3 /2F = 5, 1 was corresponded to 6 2 S1 /2F = 4 for deduction. When we apply semi-classic theory to analyze problem, the Hamiltonian could be expressed as (1) in the Schrodinger Picture. H = H 0 + H1 (1)H 1 is the free Hamiltonian without interaction between field and H 1 is the interactive Hamiltonian.When considered the Doppler Effect , the density matrix equation could be written asUnder the rotating wave approximation we can obtain newly defined density-matrix elementsThe density matrix changed into cin equations, and It has considered that the element of density matrix is different at different time point. In the above equations, it is impossible to find exactly solution. Instead, we could find a solution that is exact for an applied strong coupled field and is correct to lowest order for the probe field. So, we use transformation of element of density matrix.Insert such transformation into density matrix equations to calculate and equate with the same time dependence In order to solve equations of motion for density matrix, we use Fourier Transfer of density elementAnd we could get equations with harmony of k c,and get an infinite series of steady-state equations for the slowly varying amplitudesAnd A,B,C,R are matrixes.After careful calculation, A,B,C could be written We use mathematical method to solve the equations And the we get an infinite series solution of ( )X 5m ,then from the relationship of ( )X 5m andρ2+11 , we could getThe correspond susceptibility could be written asFor Doppler broaden media, the particles obey the Maxwell distributionThen get the dielectric and refraction indexBecause the coupling field is stand-wave field, it makes the susceptibility of media has periodical changing. So we could use transfer matrix to get transmission, reflection and absorption spectrum of probe field. When consider the media length is limited, we could set the length of the media is L = Na ( N≥1).N is the periods stack. the whole transfer is M ( N).And M is the single transfer matrix. So we could simplified asI is the unity matrix. Such a compact expression enables one to write the reflection R N and transmission TN amplitudesFrom above equations, the reflection, transmission and absorption can be readily found by calculating, respectively: R N2, TN 2and A = 1 ?RN 2?TN2. The Second part: Numerical simulation and theory analysis For the schemes of energy level, we choose parameters as T = 400K, N 0 = 5×1015 s?1γ21 = 2π×6.624MHz,Ω1 = 3γ21 is forward coupling field andΩ2=RΩ1 is backward field, which could be turned by change the reflection index R. the probe Rabi frequency isΩp = 0.03r21. The coupling field and probe field are in the same direction.When dc = 0, we simulate the condition of R=1 and R=0 separately. From the spectrum of transmission, reflection and absorption, we could get CPO induce high transmission of probe field and standing wave induce band gap.And from the Fig.2, we could get the band gap is the combination effect of reflection and absorption. Now, we use uniform Bragg fiber grating to explain the transmission and reflection spectrum under the standing wave field. For the two energy system, when the reflective index is 1. the strength of forward coupling field and backward coupling field is equal, forming to standing wave field. And we know the susceptibility of media is periodical changed. So the media forms a structure like Bragg grating, and we could use Bragg grating theory to explain the transmission and reflection: the central wavelength of probe field is close to the central wavelength of'grating'of coupling field. Corresponded to the transmission and reflection spectrum the central wavelength appears minimal value and maximal value.Then, we discuss parameters that make effect on band gap.1. Discuss band gap with different strength of coupling field.In the following Fig.3, the first line is experiment result and the second line is theory simulation result. And it could easy to know that the band gap becomes deep and the peak value of reflection become big with that the coupling field becomes 2. Discuss influence of different probe field on the band gap.The situation is similar to that of changes with coupling field, as shown in Fig.4. With more strong probe field, the value of transmission spectrum at zero detuning becomes deeper and the reflection spectrum at the same frequency detuning becomes bigger. 3. Discuss situation of transmission spectrum with different reflective index.At the begin of this thesis, we has already discussed that when the reflective index is 0 and 1, they correspond coherent hole burning and band gap. It is necessary to change the reflective index from 0 to 1 to analyze transmission spectrum. From simulation result, as shown in Fig.5, it could get process of forming band gap from hole burning with different reflective index. 4. Make a simulation of different propagation of coupling field and probe field.In the experiment figure 6, it could know that with enhancement of coupling field, the minimal value of band gap becomes little. In the simulation result, we separately choose coupling field as 9γ21,7γ21,2γ21, and we could get the deep of band gap become smaller with stronger coupling field.At last, we make a simple analysis of difference and identity of band gap between three energy level and two energy level system. No matter three energy level and two energy level system when the coupling field is standing wave field, it could both form band gap in the zero detuning frequency. However, the reflection spectrum is very different. The media become high reflective Bragg mirror because periodical refractive index, however, in the three energy level system the coupling field and probe field are interactive with different energy level, it means the probe field is detuning from central frequency of'grating'. Thus, the reflection is very week and the main reason for band gap is absorption. And we could simply think the media is a kind of absorptive grating. In the two energy system, the frequency of probe field is almost equal to the central frequency of'grating', so the reflection is strong and the media could be thought as dispersive grating.