Subluminal And Superluminal In A Four Level System

Recently, researches on subluminal and superluminal light propagationhave aroused a great deal of interest. The subliminal (or superluminal) lightpropagation is a consequence of the lossless normal (or anomalous) dispersion.In this thesis, we propose a method of realizing subluminal and superluminalby controlling the strength of the light field in a four-level system. The resultshows that the light pulse can propagate subluminally or superluminally in thesystem with zero absorption. This has a good value to the experiment work.Also,we analyzed the influences of the splitting between the upper levels onthe group speed, and compare the results between the four-level system andthe three level Λ system.As shown in Figure 1, the four-level system is structured by atomicground state |1> , middle state |2> , a pair of nearby hyper-line state 3 and|4> . State |1> and |2> are metastable. Generally speaking, |1> ←→|2> is anelectric dipole forbidden transition. The probe field E P, the control field ECand the microwave field E L drive the atomic transition |1>→|3>,|4>,|2>→|3>,|4>,and |1>→|2>, respectively. Here we analyze the model by thesemiclassical theory, under the slow varying and the rotating-waveapproximation. The three fields are treated as classical fields, as follows:E P ( r,t)= EP0 exp(?iω Pt+ikPr)E C ( r,t)= EC0 exp(?iω Ct+ikCr)E L ( r,t)= EL0 exp(?iω Lt+ikLr)Where Ω 21, Ω 31, Ω 41, Ω 32, Ω 42are the Rabi frequencies of the fieldcorresponding to transitions|2>→|1>,|3>→|1>,|4>→|1>,|3>→|2>,|4>→|2>respectively ( Ω ij =eixjEexp(?IΦ)/,i≠j,i,j=1,2,3,4),here μ ij =eixj are the induced atomic dipole moment. Choosing the phaseof light field, Φ =0, Rabi frequencies can be writen as2()()ijkijkΩ =μE(k=P,C,L).The Hamiltonian of the system is given by:Fig.1:Schematic diagram of four-level system.H = H 0 + HI (1)Where H 0 = ω1 1 1 + ω 2 2 2 + ω3 3 3 + ω44 4413242..22()33()44213141324233232213144eHcHeititI?Ω?Ω?Ω+=?Δ?Δ+Δ?Δ+Δ?Δ?Ω?Ω?Δ?Δ(2)H0 and HI represent the unperturbed and interaction parts of the Hamiltonian,respectively. Considering the decay rates of the electronics, the dynamics ofthe system is described by the density matrix equation. [ , ] 1{ , }I2ρ= ? i Hρ ? Γ ρ (3)Where Γ is a relaxation matrix In general,the relaxation process are morecomplicated., γ ij = (γ i + γj ) / 2, (i , j = 1,2,3,4, i ≠ j), γ ij (i , j = 1,2,3,4, i ≠ j)isthe decay rate respectively ,detunings Δ1 , Δ 2 , Δ 3 , Δ 4,Δ are defined as:Δ 1 =ω P ?ω31, Δ2=ωC?ω32,Δ3=ωL?ω21,Δ4=Δ1?Δ2?Δ3,Δ=ω43It is difficult to obtain analytical expressions for the set of equations. Wesolve them numerically. According to the definition of polarization:0( , ) 1[ ( , ) . ]2i ptP z t = ε E p χ z ω pe ?ω+ c c (4)Where χ (ω p )= χ′+iχ′′.The real and imaginary part of χ corresponds tothe dispersive and absorptive characteristics of the medium respectively. Byperforming a quantum average of the dipole moment over an ensemble ofhomogeneouly broadened atoms, we haveP ( z , t ) = Ω1 3 N ρ 31 + Ω1 4 N ρ 41 + c. c = 2 Ω1 3 N ρ 31 + 2Ω1 4 Nρ41 (5)where N is the atomic density. Combining Eqs (4) with Eqs(5), thesusceptibility χ and the group velocity of the probe pulse0 00Vg = Re(1 + 2π χ (ω ) + 2cπ ω ?χ? ω (ω )|ω =ω) can be obtained.In the model level |3> and |4> is very near, Δ is smaller than the decayrates respective. We can take an example of upper fine levels of the sodiumline D1 or D2 line. Generally speaking, probe field propagates in the mediumwith large absorption due to nearby hyperfine states. This is verydisadvantageous for experiments. To overcome this disadvantage, we choose amicrowave field E L in the system. For the spliting is quite larger than thedecay rates γ , the system described by Figure 1 will simplify as the threelevel Λ system of documentation [PRA,64, 053809]. In order to estimatethe influence of hyperfine level on the probe group speed, wecompare the result of three-level and this four-level model. Asshown in figure 2 and figure 3, the two systems experience zeroabsorption at the resonance frequency and group speed reduction without thefield E L.Fig.2: Imaginary parts of the susceptibility χ (ω ) as a function of the detuning Δ1 / γ, thesolid line represents the result of four-level system, parameters satisfy the following relation,the solid line is: 42 24 32 23 21 1241 14 31 135 , 5 ,, , 0.1γ γ γγ γ γΩ = Ω = Ω = Ω = Ω = Ω =Ω = Ω = Ω = Ω = Δ = , thedashed line represents the results of Λ system, parameters are used as:Ω 32 = Ω 23 = 5γ , ,Ω 31 = Ω1 3= γFig.3: Real parts of the susceptibility χ (ω ) as a function of the detuning Δ1 / γ, the solidline represents the result of four-level system, parameters satisfy the following relation, thesolid line is: 42 24 32 23 21 1241 14 31 135 , 5 ,, , 0.1γ γ γγ γ γΩ = Ω = Ω = Ω = Ω = Ω =Ω = Ω = Ω = Ω = Δ = , the dashed linerepresents the results of Λ system, parameters are used as:Ω 32 = Ω 23 = 5γ , ,Ω 31 = Ω1 3= γAs is shown in Figure 4 and Figure 5, the three level system and the fourlevel system both implement zero absorption at the resonance frequency andabnormal dispersion with microwave field E L on. The dispersion value ofthe three system is bigger than the four level model.Fig.4: Imaginary parts of the susceptibility χ (ω ) as a function of the detuning Δ1 / γ, thesolid line represents the result of four-level system, parameters satisfy the following relation,the solid line is: 42 24 32 23 21 1241 14 31 135 , 5 ,, , 0.1γ γ γγ γ γΩ = Ω = Ω = Ω = Ω = Ω =Ω = Ω = Ω = Ω = Δ = , the dashed linerepresents the results of Λ system, parameters are used as:Ω 32 = Ω 23 = 5γ , Ω 21 = Ω1 2 = 5γ ,Ω 31 = Ω1 3= γFig.5: Real parts of the susceptibility χ (ω ) as a function of the detuning Δ1 / γ, the solidline represents the result of four-level system, parameters satisfy the following relation, thesolid line is: 42 24 32 23 21 1241 14 31 135 , 5 ,, , 0.1γ γ γγ γ γΩ = Ω = Ω = Ω = Ω = Ω =Ω = Ω = Ω = Ω = Δ = , the dashed linerepresents the results of Λ system, parameters are used as:Ω 32 = Ω 23 = 5γ , Ω 21 = Ω1 2 = 5γ ,Ω 31 = Ω1 3= γIn figure.6, it is shown how the value Δ impacts the real parts of thesusceptibility χ (ω ). It is illustrated that the bigger value of Δ , the smaller ofthe index of refraction n at the zero-absorption with little change of ?χ ? ω (ω ).Fig.6: Real parts of the susceptibility χ (ω ) as a function of the detuning Δ1 / γ, the solidline represents the circumstance when the detuning is 0.64γ , while the dashed is the resultof Δ = 0.1γ, the other parameters are common:42 24 32 23 21 1241 14 31 135 , 5 , 5,γ γ γγ γΩ = Ω = Ω = Ω = Ω = Ω =Ω = Ω = Ω = Ω =In the Figure 7, the index of group velocity n g is shown versus the fieldstrength E L, the group index n g = c /vg changes from large positive valuesto large negative values and back to positive values as the intensity ofmicrowave field is increased in both three level system and four level model.Here Doppler broadening is not considered. Figure 7 indicates that our modelcan realize subluminal and superluminal light propagation as the intensity ofmicrowave field is changed.Fig.7: The index of group velocity n g as a function of the field strength E L, the solid linerepresents the circumstance when the detuning is 0.64γ , while the dashed is the result ofΔ = 3.2γ, the other parameters are common:Ω 42 = Ω 24 = 5γ , Ω 32 = Ω 23 = 5γ , Ω 41 = Ω1 4 = γ ,Ω 31 = Ω1 3= γIn conclusion, we propose on actual model and testify subluminal andsuperluminal propagation of a weak pulse light. Under the same condition, theabnormal dispersive values of four level system is smaller than three levelsystem. When splitting Δ between level |3> and |4> is quite larger than thedecay rates γ , the model described by figure 1 will simplify as the three levelΛ system. In particular, we have demonstrated how the application ofmicrowave field can produce regions of anomalous dispersion withzero-absorption and how this results in superluminal propagation of a weakpulse of light. This theoretical results will take as effect on the relativeexperiments.