Controllable Photon Band Gap With Standing Wave Drive In Cs Atomic System

This thesis is based on semi-classic theory about atomic coherence and quantum interference with method of transfer matrix, to solve Electromagnetically Induced Transparency (EIT), and controllable photon band gap with standing wave drive in Cs atomic system. In this thesis we discuss the reflection and transmission of the probe beam controlled by the standing wave to emulate experimental results. Although much work about band gap has been done before, but the influence of the Doppler effects did not been considered. In this paper, we consider the Doppler effects for hot atoms and analyze phenomena by solutions to the equations of density matrix strictly without introducing the assumptions that the atoms are laid in the ground state in the beginning.The first part: We consider theΛmodel three levels of the Cs atom, and analyze the photonic band gap which could be dynamic controllable. The atomic system is shown in Fig.I-1; in this case, the probe laser and the coupling laser are fixed on the resonant levels. In simplification, we take 1 corresponding to 6 2 S 1/2F = 3; 2 corresponding to 6 2 S1 /2F = 4; 3 corresponding to 6 2 P3 /2F = 4; the transitions 3 ? 1 , 3 ? 2are electric dipole allowed, while the transition 2 ? 1 is forbidden. We consider the coupling laser is a traveling or standing wave and consider influence of the Doppler effects for the moving atoms.In an interaction picture, the interaction Hamilton is expressed as follows in the Hilbert space spanned by the bare states 1 , 2 , 3 with the rotational wave approximation:The atomic system is shown in Fig.I-1,Δp =ωp?ω31andΔc =ωc?ω32 which are the detunings of the probe and the strong coupling waves.In the same basis the Hamiltonian of a free atom is written as:The response of the macroscopic medium to the field is governed by the density-matrix equation:Considering the moving of the atoms, we have:Under the rotating wave approximation we can obtain newly defined density-matrix elementsInsert(I.2),(I.3),(I.6)into(I.5), the following equations of motion for the matrix elements are obtained:Where we chooseIn this paper, the standing wave consists of a forward and a backward traveling wave:In order to solve(I.7),we introduce column vector ofψThe steady-state solution is obtained as a spatial Fourier seriesWe inser(tI.8),(I.10)into(I.7):Equate the coefficients of harmonics of k c, and obtain an infinite series of steady-state equations for the slowly varying amplitudes: Where A,B,C,R are matrixes.After we got the A, B and C for the system in fig. I.1, as follows: Arrange the slowly varying amplitudes into a column vectorThe steady-state solution for the slowly varying amplitude is derived as WhereTherefore we can get the value of X, that mean we can getψwhich stand for the density-matrix operator, which are correlated with the distance z , but timet . To consider the atomic motion, we need integrate the value ofρ31( m)to all responses of atoms with different velocities by Maxwell distribution.The relevant susceptibility raftered by the probe field can be written as: Where N ( v ) is the function of Maxwell velocity distributionWe can get the dielectric:This results from that Rabi frequency of driving field varies periodically along x, with a spatial periodicity a =λc2.According to Bloch's theoremWhere I is the unity matrix. Such a compact expression enables one to write the reflection R N and transmission TN amplitudes for an N periods stack in terms of the complex Bloch wave vector k and the elements mi j of the matrix M, namely, From which, in turn, the reflection, transmission and absorption can be readily found by calculating, respectively: R N2, TN 2and A = 1 ?RN 2?TN2.The second part:Experimental results and theoretical simulation.Corresponding to the energy level sketch shown in figI.2a and I.2b,experimental results are obtained as shown in fig I.3a and I.3b : FigI.3a and FigI.3b are the detection of the transition signal which is varied by changing the detuning ofΔp .The parameter to simulation the EIT and PBG, where the length of the mediu L = 2.5cm,ωc =3.517217095386006×101 4×2π,ωp=3.517217095386006×101 4×2π,Δc = 0,γ31 =γ32=6.234×10 6×2π,γ21=250×10 3×2π,Δp = ?5 0γ31 ??5 0γ31,Ω? = R×Ω?. The Rabi frequency of forward travelling wave is denoted asΩ1 ,the backward isΩ2=RΩ1 . We can control the value of R to simulation the standing wave or traveling wave for coupling field,the intensity of coupling wave is larger than the probe laser,where we takeΩp =0. 001r31,and they are in the same direction.WhenΔc= 0,we take the R=0,R=1 to simulation the cases of traveling wave and standing wave,where we takeΩ1 = 5γ31. We simulate the transmission ,reflection and absorption,and found that phenomena of EIT and band gap is observed as FigI.4. WhenΔp= 0,the R changed from 0 to 1,the transmission is became smaller,the band gap is formed,whenΩ2 =Ω1 ,the transmission is changed to zero.We simulation the transmission ,reflection and absorption for the conditions that the value of Re is R=0.5,R=0.6 and R=0.8 as shown in FigI.5,from the picture,we can get the conclusions that as the R is get larger,the band gap is formed gradually. Because the condition of Bragg is not satisfied in our experiment and simulation, the reflection is very small, so we can neglect the influence of reflection in our discussion.We also did the experiments in the atomic system as figI.6.In order to get the results of the band gap changing as the vary the intense of the coupling wave,we get the results of FigI.7, which the coupling wave intense is changedAnd we get the results as shown in I.8a,b,c, which corresponding to the transmission of traveling wave, and the transmission and reflection in the case of standing wave. We take the same parameters as in FigI.2 and For the conditions that R=0 and R=1,we obtain the transmission, reflection and absorption curves as FigI.9.For the conditions that R=0.5,R=0.6 and R=0.8,we obtain the transmission, reflection and absorption curves as FigI.10. Here we obtained the band gap which varies with the intense of the coupling wave as shown in FigI.11, which could explain the phenomenon of the results as FigI.7. In the above experiments and simulations, we find that the width of the band gap could be changed with the intense of coupling, the frequency detuning of coupling wave. It is different from that by Bragg gating, for which the band gap is fixed and could not be changed. We can get band gap dynamically controlled.The third part:The reasons that the band gap formedConsidering the frequency detuning between the levels of 6 2 S1 /2 F = 3→6 2P1 /2F= 4 is 9.1GHz,and Bragg condition is not satisfied in our experiment and simulation, so the reflection is very small. We can neglect the influence of reflection in our discussion, the band gap is caused mainly be the absorption.We can also to simulate the experiment in another way which illuminates the phenomenon indirect by the absorption:The intensity absorption for the probe is given by the image ofχ:If the length of the active is L , and insert n nikczσ31e2into (I.21), then we have absorption coefficient:Where 2λ=πkcis the wavelength of the coupling standing-wave, sinceλL<< 1, the second term could be negligible, then we have:We can get the absorption character of the probe wave as shown in fig.12.The PBG is caused mostly by the absorption, we simulation the absorption when the R is in different value, we get the fig.I-12, form which we can see that when the R=0, near the scope ofΔp = 0, it satisfy the condition of EIT, the absorption is small, resulting in a big transition, but when the R is becoming to one, at the same time, the SW is forming gradually, we find that the absorption is get more and more bigger, which the phenomenon is named EIA, resulting in the transition is becoming smaller, these accord with our experiment results very good, so we can say the PBG is caused by EIA.For the case that satisfied the condition of Bragg condition, as the paper showed,which satisfied the Bragg condition by control a small angle where photonic band gap is caused by both absorption and reflection together. We did experiments in different condition compare to them.In conclusion,we have studied in dynamic controllable photon gap with standing wave drive in hot Cs atomic system. We obtained the experiment results of EIT and band gap, and simulated these experimental results and explained the reasons that results in those in several cases. Here the band gap could be dynamically controlled by adjusting the parameters such as intensity of the coupling wave, and the reflection coefficient of the mirror and other complications...