Transmission and reflection of electromagnetically induced absorption grating in homogeneous atomic media

We theoretically study the transmission and reflection of the probe travelling wave in an electromagnetically induced absorption grating (EIG), which is created in a three-level Λ-type atomic system when the coupling field is a standing wave. Using the system, we show that a photonic stop band can exist on one side away from the resonance point in ultracold atomic gas, while there is an enhanced absorption at resonance and small reflection around it in the thermal atomic gas. Because our method can deal with such two cases, it is helpful to further understand the effects of the Doppler effect on atomic coherence and interference. © 2008 Optical Society of America OCIS codes: (270.1670) Quantum optics: Coherent optics effects; (020.1670) Atomic and molecular physics: Coherent optical effects; (050.2770) Diffraction and gratings: Gratings. References and links 1. H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. 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Introduction
Recently, the three-level Λ-type atomic system under the condition of electromagnetically induced absorption grating (EIG) where the coupling wave is a standing wave has attracted great attention [1][2][3][4][5].The calculation of the diffraction of a weak probe field which propagates along a direction normal to the standing wave was reported [1], and the related phenomenon was observed in atomic ultracold gas [2].When the probe wave propagates along the direction of the standing wave, the transmission and reflection of the probe field become very interesting and useful.M. Artoni et al theoretically presented a scheme to realize optically tunable photonic band gap in ultracold atomic gas [3], while Brown et al formed an all-optical two-port optical switch in atomic thermal gas [4].At first, it is noted that Artoni realized his assumption in ultracold atomic gas while Brown made the experiments in a thermal gas media, but their results are different.For example, at the resonance frequency, there is a transmission of the probe field from Artoni's results, while there is an enhanced absorption from Brown's.Secondly, it is noted that the experiment of Bajcsy et al [6] is similar with EIG, but it is a different physical process.The experiment of Bajcsy is based on the theory of dark-state polariton [7,8], and Doppler shifts due to atomic motions do not affect the signal beams which are converted from the atomic spin coherence [9].However, in the case of EIG, the reflected beam comes from the probe wave, the Doppler shifts have a great effect on the signal beams.And so far the existing theoretical calculations [3,6,[9][10][11][12][13] which have been performed for related standing wave system can not be directly applied to explain the experimental results of Brown.In this paper, we theoretically present a numerical method to describe the behaviors of the probe wave in the case of EIG when the probe wave propagates along the direction of the standing wave.In the limit with an assumption of zero atomic velocity, our numerical simulation has a good agreement with the results of Artoni.Considering the Doppler shifts due to atomic mo-tion, we give a corresponding mathematic calculation for the experiments carried by Brown.For the reason that our method are suitable for the both cases, it is useful to further understand the effects of Doppler effect on atomic coherence and interference.

The model and equations
The atomic system under consideration is shown in Fig. 1(a).It can be described by the three level Λ-type configuration in a closed system.A weak probe wave E p (z,t) = where g = μ 31 ε p /2h, Ω 1 = μ 32 ε 1 /2h and Ω 2 = μ 32 ε 2 /2h are the Rabi frequencies of the probe, co-propagating and counter-propagating waves, respectively, and here μ 31 and μ 32 are electric dipole moments between the corresponding states.Without loss of generality, the three Rabi frequencies can be taken to be real.Including the relaxation terms of the system, which can be described by means of a Liouville operator R, the evolution of the density matrix ρ(υ, z,t) is given by [14,15] ∂ ∂t where υ is the atomic velocity along the z-axis.Due to the high relative population of the level |1 induced by the strong standing wave, we can ignore all effects which are related to the changes in population and coherence at the transition |3 -|2 .So we focus the spectroscopic effect in such system to the off-diagonal elements ρ 31 and ρ 21 , and set ρ 11 = 1, ρ 22 = ρ 33 = 0.At the stationary state, the density-matrix equations reduce to [16] where , the terms γ i j describe the coherence dephasing rates of the off-diagonal elements.In order to solve Eq.( 3), we express the elements of the density matrix as With inserting of Eq.( 4) into Eq.(3), we eliminate σ 21 (n) and lead to an inhomogeneous difference equation here the coefficients are and We solve the elements in Eq.( 5) by the method of continued fractions [17,18].To consider the atomic motion, we integrate the value of σ 31 (n) to all responses of atoms with different velocities by Maxwellian distribution. where 2RT /M represents the most probable atomic velocity.After considering the motion of atoms, we can solve the system as a homogeneous medium.And the relevant macroscopic susceptibility and dressed dielectric function experienced by the probe can be written as here N = (λ p /2π) 3 N 0 presents the scaled average atomic density, and λ p is the wavelength of the probe wave, N 0 is the atomic density.
Owing to the spatially periodic modulation induced by the standing wave, the weak probe wave propagates as in a one-dimensional grating with periodicity a = λ c /2, which is the half of wavelength of the coupling wave.In order to characterize the structure, we numerically calculate the 2 × 2 unimodular tranfer matrix M(Δ p ) which represents the propagation of the probe wave through a single period of length a [19].Then, the translational invariance of the periodic medium is fulfilled by imposing the Bloch condition on the photonic eigenstates, where E + and E − are the electric field amplitude of the forward and backward (Bragg reflected) propagating probe, respectively, and κ = κ + iκ is the Bolch complex wave vector of the corresponding probe photonic states.The one-dimensional grating structure is obtain from the solution of the corresponding determinantal equation e 2iκa − Tr[M(Δ p )]e iκa + 1 = 0 (detM = 1) and we note both κ and −κ are the solutions of the equation.We can simply has In real experiments, we pay attention to the propagation through samples of finite length.So calculating the corresponding reflectivity and transitivity spectra, it is necessary to consider a finite sample of thickness L = Na, where N is the number of the standing wave periods.The total transfer matrix M (N) of a sample with thickness L = Na (N 1) is given in term of the single period transfer matrix M as M (N) = M N .Because M is unimodular, the following closed expression for M (N) holes true here I is the unity matrix.With the compact expression, we are able to calculate the reflection (R N ) and transmission (T N ) amplitudes for the length L in terms of complex Bolch wave vector κ and the element M i j of the matrix M from Eq.( 12), the reflectivity, transmissivity and absorption can be found by calculating

Numerical results and discussion
Based on the above formulas in the previous section, we consider the following three cases of EIG: (i) The probe wave propagates along the direction of the standing wave inside a ultracold atomic gas, as shown in Fig. 1(b).
(ii) The probe wave propagates along the direction of the standing wave inside a thermal atomic vapor, as shown in Fig. 1

(b).
(iii) There is a small cross-angle between the probe wave and the standing wave inside the thermal atomic vapor, as shown in Fig. 1(c).
For the three cases, we choose the typical parameters for the D1 line of 87 Rb at 795nm, γ/2π=1.5MHzand γ 31 = 2γ.In the configuration of case (i), we assume γ 21 =0.001γ, the atomic density N 0 = 10 12 cm −3 and the length of the sample L=2.5cm.For the ultracold atomic medium, we also assume υ=0m/s in Eq.( 5) and display the reflectivity and transmissivity of the probe wave in Fig. 2(a).An investigation of Fig. 2(a) shows that in the non-perfect standing wave configuration Ω 2 = 0.8Ω 1 , one can obtain a photonic stop band on one side away from the resonance point.This phenomenon is caused by the spatially periodic modulation induced by the standing wave.Owing to the spatial modulation, the absorption of the probe wave is modulated spatially in the same period.And because the real and imaginary parts of the dielectric function ε(Δ p ) in Eq.( 8) are related by the Kramers-Kronig relation [20], the periodical modulation of absorption causes a modulation of dispersion, a one-dimensional grating is formed as shown in Fig. 3(b).This grating induces the reflectivity of the probe wave, and even forms a photonic stop band.This result has been reported by Artoni [3] who assumed the Rabi frequency of the standing wave in a cosine form Ω 2 c (z) = Ω 2 1 + Ω 2 2 + 2Ω 1 Ω 2 cos 2k c z, while we expand the element ρ 31 in a Fourier series in Eq.( 4).In the case (i), our results have a very good agreement with that of Artoni as shown in Fig. 2(a), and it is a good accuracy to sum the spatial harmonics from n=-20 to 20 in Eq. (7).
For the case (ii), we choose the parameters as γ 21 =0.02γ, the atomic density N 0 =10 10 cm −3 , and the length of the medium L=7.5cm.Considering the atomic motion, we assume υ p =250m/s The results of Artoni with the same parameters.(red dash line).(b) The transmissivity of the probe wave of case (ii) with only the co-propagating wave, Ω 1 = 10γ, Δ c = 0 (dash line), and with perfect standing wave Ω 1 = Ω 2 = 10γ, Δ c = 0 (solid line).The inset shows the dressed state picture of the three-level Λ-type system coupled by a bichromatic wave and a weak probe wave, and δ is the half frequency between the two coupling waves. in Eq.( 7), and present the results in Fig. 2(b).An investigation of Fig. 2(b) shows that we can obtain a narrow transparency window at the resonance point in the presence of only the copropagating coupling wave, and this is the character of EIT [21].And as the standing wave is formed, the transparency window of EIT changes to an enhanced absorption and two transparency windows emerge around the resonance point.This result can be understood by investigating the elements σ 31 (0, υ) f (υ) which are the responses of the atoms with a certain velocity shown as insets in Fig. 3(a).In the atomic rest frame, the coupling wave is a standing wave for the stationary atoms, and such configuration is the same as that of case (i).While for the atoms with a velocity υ, the standing wave should be decomposed into a pair of travelling wave components, the frequency of the forward wave shifts to ω c + k c υ and that of the backward wave shifts to ω c − k c υ.So the configuration of atoms with a velocity can be equal to the situation that the stationary atoms are pumped by a bichromatic wave, and the two coupling waves have a frequency difference 2δ = 2k c υ.In the dressed state of the stationary atoms driven by a bichromatic wave [22][23][24], the excited state |3 is decomposed into a periodical structure |m with a separation δ shown as inset in Fig. 2(b).So the resonances of absorption can be located at Δ p = k p υ ± k c υ, k p υ ± 2k c υ, k p υ ± 3k c υ... which are consistent with the results shown as insets in Fig. 3(a).Due to k c ≈ k p , one can have resonance of absorption at Δ p = 0 for the atoms with an arbitrary velocity.So after considering the responses of all atoms with different velocities, we obtain the value of Ψ 31 (0) in Fig. 3(a) which is consistent with the result of EIG in Fig. 2(b).In the situation of case (ii), due to the Doppler shifts and the small atomic density, although the grating induced by the standing wave is formed, it becomes blurry spatially and the modulation of the refractive index is too small as shown in Fig. 3(b).As a result, the Bragg condition nλ c = λ p (λ c =794.983nm, λ p =794.968nm) can not be satisfied, and there are nearly no reflection signals.
In the following, we pay attention to the case (iii).And the only difference between the case (iii) and the case (ii) is that there is a small propagation angle θ between the probe wave and the standing wave.Developing the theory, we set the wave vector of the probe wave as k p = k p cos θ z + k p sin θ x, and set k p • υ = k p cosθ υ z + k p sin θ υ x in Eq.( 6) [25].For simplicity, we choose θ = arccos(λ p /λ c ) in order to satisfy the Bragg condition λ p = nλ c cos θ .Due to the angle is rather small, as a good approximation, we ignore the motion of atoms along axis x and set k p • υ ≈ k p cosθ υ z .For the case (iii), we replace the refractive index n by p = n cosθ in Eq.( 8) [26] and supply the corresponding results in Fig.The spectral characters of Fig. 4 have a good agreement with the experimental results reported by Brown et al [4,5], although there is a discrepancy that the small reflectivity at the resonance point in our calculation.The discrepancy is probably due to the reasons: (i) our calculations do not consider the broadening effects of the finite laser linewidth.(ii) we ignore the zeeman broadening from the residual magnetic field in the experiment.(iii) we do not consider the contributions from a nonlinear index grating [27,28] which depends the intensity of the probe wave.However, we believe that our numerical method gives a physical understanding and qualitative features of the experiment of Brown.

Conclusion
In conclusion, we theoretically present a numerical method to solve the transmission and reflection of the probe wave in the condition of EIG in a three-level Λ-type atomic system driven by a standing wave.Our theoretical method is not only suitable for the sample of ultracold atomic gas but also useful for the Doppler-broadened atomic gas.For the system of ultracold atoms, the results of our method are consistent with the reports of Artoni et al; for Doppler-broadened atomic system, our results are agree with the experimental results of Brown et al, and it is the first time to understand their experimental results with calculations.For this reason, our method is a way for the understanding of the effect of the Doppler effect on atomic coherence and interference.

Fig. 1 .
Fig. 1.(a) Schematic diagram of the three-level Λ-type atomic system driven by the strong standing wave and probed by the weak wave.(b) A block diagram of the case (i) or (ii).(c) A block diagram of the case (iii).

Fig. 2 .
Fig. 2. (a) (Color online) The reflectivity of the probe wave and the transmissivity as inset in nonperfect standing wave configuration of case (i), Ω 1 = 10γ,Ω 2 /Ω 1 = 0.8 and Δ c = 0 (black solid line).The results of Artoni with the same parameters.(red dash line).(b) The transmissivity of the probe wave of case (ii) with only the co-propagating wave, Ω 1 = 10γ, Δ c = 0 (dash line), and with perfect standing wave Ω 1 = Ω 2 = 10γ, Δ c = 0 (solid line).The inset shows the dressed state picture of the three-level Λ-type system coupled by a bichromatic wave and a weak probe wave, and δ is the half frequency between the two coupling waves.

Fig. 3 .
Fig. 3. (a) (Color online) The element Ψ 31 (0) for the responses of all atoms with different velocities.The insets show the elements σ 31 (0,υ) f (υ) for the response of the atoms with a certain velocity.The parameters are taken to be the same as in Fig.2(b).(b) The refractive index of case (i) (dash line).Δ p = −0.1γ,other parameters are the same as in Fig.2(a).And the refractive index of case (ii) (solid line).Δ p = 7γ, other parameters are the same as in Fig.2(b).For comparison, the value is multiplied by 100.

4 .
An investigation of Fig.4(b)  shows that the EIG induced reflection signal has two peaks when the standing wave is resonance, and when the detuning |Δ c | > 20γ, one can obtain the reflection signal with one peak.This phenomenon can be understood that because the Bragg condition is satisfied in case (iii), the effect of the EIG induced grating emerges, the reflection signal is enhanced.And due to the grating is caused by the spatial modulation of absorption, the reflection signals should locate at the positions having transparency of the element Ψ 31 (0).The two dips of the transparency of the probe wave as shown in Fig.4(a) are consistent with the two peaks of the reflection signal.When the standing wave has detunings, the positions of the reflection signal change with that of the transparency windows.And if |Δ c | > 20γ, one can obtain the reflection signal with one peak because there is only one transparency window.

Fig. 4 .
Fig. 4. (a) The transmissivity of the probe wave in case (iii), Δ c = 0. (b) (Color online) The reflectivity of the probe wave as a function of Δ p for different detunings Δ c of the standing wave in case (iii).Other parameters are taken to be the same as in Fig. 2(b).