Recently, the research of light-media interaction has played a significant role in the area of quantum optics, where many interesting optical phenomena can be caused by atomic coherence and quantum interference. Of them, the electromagnetically induced transparency (EIT) is the most typical.^[1,2] The EIT is mainly used to manipulate the refractive and dispersive properties by suppressing or eliminating the absorption for a weak probe beam in atomic medium driven by an additional strong coherent field. Meanwhile, coherent population trapping (CPT) state,^[3,4] or the so-called dark state, can be formed in this process. The principal mechanisms for EIT and CPT have been explored in many recent applications, such as lasing without inversion,^[5,6] refractive index enhancement,^[7,8] slow light,^[9,10] light storage,^[11,12] enhanced Kerr nonlinearity,^[13,14] optical solitons,^[15,16] etc.

As is well known, an electromagnetically induced grating (EIG) can be created in a medium by replacing the travelling-wave driving field in EIT with a standing-wave. The probe field will obtain a periodic variation of absorptive coefficient due to the spatial modulation induced by the standing-wave field. At the same time, the refractive index also changes periodically.^[17] Then the medium is equivalent to a grating on which the resonant probe beam can be diffracted into high-order directions. This phenomenon has been observed experimentally in cold and hot atomic samples^[18,19] and universally applied to optical bistability,^[20] all optical switching and routing,^[21] beam splitting and fanning,^[22] quantum Talbot effect,^[23,24] surface solitons,^[25] photonic Floquet topological insulators,^[26] etc. However, the diffraction efficiency of EIG is not enough due to the inevitable atomic absorption. Therefore, an electromagnetically induced phase grating (EIPG) was proposed for high diffraction efficiency by using enhanced Kerr nonlinear refractivity at low light level.^[27,28] Likewise, one can build gratings in other media or configurations such as quantum well,^[29] quantum dot,^[30] molecular magnets,^[31] double-dark resonances system,^[32,33] phase-dependent system^[34] and CPT system.^[35] Meanwhile, a kind of gain-phase or gain grating based on the effect of active Raman gain (ARG) has been put forward to obtain much higher diffraction efficiency in high-order directions.^[36] Recently, Wang et al. extended EIG to the two-dimensional (2D) case and achieved an electromagnetically induced cross-grating (EICG) in a four-level tripod-type atomic system.^[37] Then, schemes for EICG based on EIT or ARG in multi-level atomic systems were proposed, and the diffraction efficiencies were improved.^[38–41]

In this paper, we theoretically investigate a one-dimensional (1D) phase grating coherently induced in a mixture of two species of Λ-type atomic system with EIT and ARG, respectively. This three-level atomic system has been proposed for the enhancement of refractivity without absorption.^[42,43] Recently, the creation of a spatially distributed PT-symmetric refractive index was explored.^[44] Here, by using two far-detuned standing-wave fields to drive the corresponding atomic transitions in the vicinity of two-photon resonance, we obtain spatial absorption-phase, pure-phase, and gain-phase gratings through modulating the interaction of the two Λ schemes. Thus, the probe beam can be diffracted to high-order directions with different intensities. It is found that the diffraction efficiency of the grating depends on the detunings of the two two-photon Raman transitions and the amplitudes of the standing-wave fields. We also analyze the physical mechanism for such a mixture of EIT and ARG in a real atomic system with ⁸⁵Rb and ⁸⁷Rb. The scheme can be used as all-optical switching and beam splitter which have potential applications in optical networking and communication.

The considered atomic system is a mixture of two three-level Λ-type atomic species (EIT and ARG) interacting with three coherent laser fields as illustrated in Fig. 1. The atomic densities of the species are N₁ and N₂, respectively. Then the total atomic density is N = N₁ + N₂. Each of Λ systems is marked by si (i = 1,2) with one excited state |3,si〉 and two ground levels |1,si〉 and |2,si〉. A weak probe field with frequency ω_p and a pair of coupling fields with frequencies ω_s1 and ω_s2 drive the corresponding transitions. The Rabi frequencies of the driving fields are denoted as Ω_si = μ_lm,siE_si/2ħ and Ω_pi = μ_lm,siE_p/2ħ (i = 1,2), where E_si and E_p represent the amplitudes of electric fields and μ_lm,si (l, m = 1, 2, 3) are the dipole moments of the corresponding transitions. We define that Δ_s1 = ω_31,s1 − ω_s1 and Δ_s2 = ω_32,s2 − ω_s2 are the one-photon detunings for the two coupling fields, and Δ_p1 = ω_32,s1 − ω_p and Δ_p2 = ω_31,s2 − ω_p are the detunings for the probe field with ω_lm,si(l,m = 1,2,3; i = 1,2) being the resonant frequencies of corresponding transition in si-system. Here, all the involved fields are detuned far from the excited state |3,si〉. For the EIT system, the atom is transparent to the probe field under the condition of two-photon resonance, i.e., δ₁ = 0, and its refractive index is unity. However, one can obtain a narrow absorption peak near the two-photon resonance.^[45] At the same time, the refractivity is enhanced due to the Raman process. For the ARG system, the probe field can be amplified by the atoms and a nonzero susceptibility is then achieved when the two-photon detuning δ₂ ≠ 0. The combination of Raman absorption and Raman gain can result in enhanced refractivity with vanishing absorption. The laser driven ⁸⁵Rb and ⁸⁷Rb atoms can be utilized as the two three-level Λ systems, of which one provides an absorption resonance and the other induces a gain resonance for the probe beam.^[42,43]

Fig. 1.

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	Fig. 1. (color online) Energy level scheme based on a mixture of two species of Λ atomic system driven by three light fields. The initially populated levels are indicated by the dots.

In the framework of the semiclassical theory, with the rotating-wave approximation and the electric-dipole approximation, the Hamiltonian of si-system in the interaction picture can be written as 1a 1b In Eq. (1), the quantities δ₁ and δ₂ are the two-photon detunings of the laser beams from each Raman transition, and they are defined as δ₁ = ω_s1 − ω_21,s1 − ω_p and δ₂ = ω_s2 + ω_21,s2 − ω_p, respectively.

By using the Liouville equation, we obtain the relevant density-matrix equations as follows:2a 2b 2c 2d 2e 2f with and . In the above equations, we consider the relaxation process, and γ_lm,si represents the dephasing rates of |l,si〉 ↔ |m,si〉 transitions. We assume that the atoms are initially populated in one of the two ground-state sublevels, i.e., ρ_22,si = 1(i = 1,2).

By solving Eqs. (2a)–(2f) in the steady state, the coherent terms corresponding to the probe transitions can be derived as follows: 3a 3b

In this system, the refractive index enhancement of the medium relies on the interaction strength between two Raman transitions, the first of which results in two-photon absorption and the second one contributes two-photon gain to the probe field. Therefore, the total complex susceptibility is equal to the sum of individual contributions from each of the Λ-type schemes, and the absorption and refractivity can be controlled by changing the two-photon detunings δ₁ and δ₂. According to the polarization of the medium, P_p = ε₀χE_p = N₁μ_23,s1ρ_32,s1 + N₂μ_13,s2ρ_31,s2, the susceptibility of the weak probe field can be written as4 where ε₀ is dielectric constant in a vacuum.

Therefore, the susceptibility of the whole system is given by 5

In order to investigate the coherently induced grating in refractive index enhanced medium, we use two standing-wave fields Ω_s1(x) = Ω_s2(x) = Ω_ssin(πx/Λ) to drive the atoms, with Λ being the spatial period, and assume that the probe field propagates along the z direction through an atomic medium with length L. The propagation of probe field determined by the atomic polarization, which, in the slowly varying envelope approximation and the steady-state regime, can be described by the Maxwell’s equation as6 where λ_p is the wavelength of the probe field. By solving Eq. (6), the transmission function of the probe field at z = L can be derived as 7 The terms of α(x) = πIm[χ]L/λ_p and ϕ(x) = πRe[χ]L/λ_p represent the absorption coefficient and phase shift of the probe field, respectively. According to the Fraunhofer diffraction theory, we obtain the Fraunhofer-diffraction intensity as 8 where corresponds to the Fraunhofer diffraction of single spatial period, θ is the diffraction angle with respect to the z direction, and M is the number of spatial periods of the grating. Through Eq. (8) with grating equation sin θ = nλ_p/Λ, which determines the n-th order diffraction angle, we can calculate the zeroth-order, first-order and second-order diffraction intensities as follows: 9a 9b 9c

In the following, we focus on a real system and consider two Λ schemes in a dual magneto-optical trap of rubidium atomic isotopes with ⁸⁵Rb (for EIT) and ⁸⁷Rb (for ARG) using the D₁ line (λ_p = 795 nm), and assign |1,s1〉 = |5²S_1/2, F = 2〉, |2,s1〉 = |5²S_1/2, F = 3〉, and |3,s1〉 = |5²P_1/2, F = 3〉 for s1-system, and |1,s2〉 = |5²S_1/2, F = 1〉, |2,s2〉 = |5²S_1/2, F = 2〉, and |3,s2〉 = |5²P_1/2, F = 1〉 for s2-system. In the following calculations, the parameters of this atomic system are N₁ = N₂ = 5.0 × 10¹² cm⁻³, μ_23,s1 = μ_13,s2 = 2.5377 × 10⁻²⁹ C · m, γ = γ_32,s1 = γ_31,s2 = γ_32,s2 = 5.75 MHz/2, and γ^′ = γ_12,s1 = γ_12,s2 = 0.05γ. We can manipulate the susceptibility by changing the two-photon detunings δ₁ and δ₂ to obtain diverse phase gratings, and the variations of intensities of driving fields can be used for achieving different phase shifts to enhance the diffraction efficiency.

In order to acquire a high diffraction efficiency for probe beam with the coherently induced grating, we must pay attention to two factors: a large phase shift should be firstly possessed across the probe beam, and the transmission of the probe must be large enough. By using the equations as outlined above, we investigate the formation of phase grating and its diffracting power in this section, and discuss the Fraunhofer diffraction patterns of the probe field in three types of gratings with different phase shifts.

The susceptibility of the medium plays a significant role in the formation of EIG, and we firstly focus on the probe absorption Im(χ) and dispersion Re(χ). As discussed above, we can efficiently modulate the strength of the two-photon Raman absorption and Raman gain via manipulating the two-photon detunings δ₁ and δ₂ of the laser beams, so that the absorption can be reduced and the probe can even be amplified. At the same time, the real part of the susceptibility also obtains a large value arising from the constructive contributions of two Raman transitions. In addition, the separation value δ₁ − δ₂ of two Raman resonances determines the interaction between EIT and ARG systems, and then the refractive index is dependent on the separation of Raman resonances.^[42,43] For δ₁ − δ₂ ≫ γ^′, two Raman transitions are independent of each other and Re(χ) drops. But for δ₁ − δ₂ ≪ γ^′, the effect of two Raman transitions can mutually offset. In such a case, both the absorption and the refraction are cancelled. Therefore, we choose that the separation value of two Raman resonance satisfies δ₁ − δ₂ = 3γ^′ here, which is of the order of γ^′.^[42,43] As shown in Fig. 2, the dispersive part Re(χ) and absorptive part Im(χ) of the probe susceptibility are plotted separately as a function of two-photon detuning δ₁ for δ₁ − δ₂ = 3γ^′, marked with the blue-solid line and red-dashed line, respectively. We take the Rabi frequencies of two driven fields as Ω_s1 = Ω_s2 = 4γ. We consider three different conditions for which δ₁ = 0.5γ^′ (point A), δ₁ = 1.5γ^′ (point B), and δ₁ = 2.5γ^′ (point C), respectively. At point B, the absorptive part of susceptibility [Im(χ)] vanishes due to the balance between two-photon absorption in EIT system and two-photon gain in ARG system while the dispersive part [Re(χ)] is enhanced due to their constructive contributions. For δ₁ < 1.5γ^′, the atomic medium is absorptive to probe beam. In this case, the slight deviation from the EIT resonance results in an absorption which is stronger than the gain provided by ARG. However, under the condition of δ₁ > 1.5γ^′, the Raman gain is dominant and the probe light can be amplified by the atoms. Nevertheless, Re(χ) is always enhanced, that is, the atomic system is a refractive index enhanced medium (see points A and C).

Fig. 2.

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	Fig. 2. (color online) Real (solid curve) and imaginary (dash curve) parts of the probe susceptibility versus the two-photon detuning δ1 under the condition of δ1 − δ2 = 3γ′. Parameters are N1 = N2 = 5.0 × 1012 cm−3,γ = γ32,s1 = γ31,s2 = γ32,s2 = 5.75/2 MHz, γ′ = γ12,s1 = γ12,s2 = 0.05γ, Δs1 = −862γ, Δs2 = 862γ, and Ωs1 = Ωs2 = 4γ.

The strength of the two-photon absorption in EIT system and the two-photon gain in ARG system are dependent on the amplitudes of the coupling fields [see Eq. (5)]. Then coupling fields with standing-wave pattern can result in a spatially modulated grating for the probe fields. As shown in Fig. 3, the properties of grating are different for the probe field when two-photon detunings of the two Raman transitions δ₁ and δ₂ take different values. With the parameters fixed, a large phase modulation is observed, reaching a peak phase-shift value of π at x = ±0.5Λ for Ω_s = 4.5γ [see the solid lines in Figs. 3(a1)–3(a3)]. However, the amplitude modulations vary with different two-photon detunings [see the dashed lines in Figs. 3(a1)–3(a3)]. Figure 3(a1) shows the transmission function for the two-photon detunings of δ₁ = 0.5γ^′ and δ₂ = −2.5γ^′. There is a certain amplitude modulation of transmission function, which oscillates around an average transmissivity of 46.8% along the x direction in a period of Λ. Therefore, the probe field has a significant energy loss inside the atomic medium due to the absorption effect in the EIT system with non-zero two-photon detuning. This is a limit to the diffracting power of the atomic grating. The resulting first-order diffraction efficiency of grating reaches only to 8.9% as shown in Fig. 3(a2). Thus, this grating is a mixture of an absorption grating and a phase grating. Figure 3(b1) shows the transmission function for δ₁ = 1.5γ^′ and δ₂ = −1.5γ^′. We can obtain a phase modulation of the order of π for the probe beam with extremely low energy loss in the atomic medium, which is due to a good balance between the two-photon absorption in EIT system and the two-photon Raman gain in ARG system. As a result, a pure phase grating with nearly 100% transmissivity is developed in the medium and the diffraction efficiency of probe beam is improved. The energy diffracted into the first-order direction can reach to 31.5%, which means that this EIG is very close to an ideal sinusoidal phase grating (about 34%) as shown in Fig. 3(b2). In the same way, we plot the transmission function with the detunings being δ₁ = 2.5γ^′ and δ₂ = −0.5γ^′ in Fig. 3(c1). It shows that the amplitude modulation of transmitted probe light is periodically modulated in space with an average transmissivity large than 100% which corresponds to an amplification of the probe beam. At the same time, the resulting phase modulation within a period is still π. Here, the probe gain is attributed to the effect of ARG. The far-field diffraction pattern of this grating is depicted in Fig. 3(c2). It is clear that the first-order diffraction efficiency can exceed 100%. Therefore, a kind of gain-phase grating is obtained. From Figs. 3(a2), 3(b2), and 3(c2), we can observe that a small amount of energy is also shifted into the second diffraction orders which are located at sin θ = ±0.5, and the diffraction efficiency is enhanced with the increase of total transmittance.

Fig. 3.

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Fig. 3. (color online) Amplitude |T(x)| (red dashed line) and phase ϕ(x) (blue solid line) of the transmission function T(x) plotted over two space periods for different two-photon detunings: (a1) δ1 = 0.5γ′, δ2 = −2.5γ′; (b1) δ1 = 1.5γ′, δ2 = −1.5γ′; (c1) δ1 = 2.5γ′, δ2 = −0.5γ′. ((a2), (b2), and (c2)): Plots of diffraction pattern I(θ) versus sin θ corresponding to the (a1), (b1), and (c1) cases. Parameters are Ωs = 4.5γ, L = 3 mm, M = 6, and Λ = 4λp. Other parameters are the same as those in Fig. 2.

The diffraction efficiency depends on the two-photon detunings δ₁ and δ₂ and the intensity of driving standing-wave field Ω_s. Figure 4 displays the changes of transmission function and diffraction pattern for Ω_s = 5.8γ. Other parameters are the same as those in Fig. 3. It is obvious that we can obtain a phase modulation of the order of 1.5π at x = ±0.5Λ [see the solid lines of Figs. 4(a1), 4(b1), and 4(c1)]. However, the amplitude modulation |T(x)| varies with two-photon detuning. For the cases of δ₁ = 0.5γ^′ and δ₂ = −2.5γ^′, the amplitude modulation is plotted in Fig. 4(a1) with the dashed line. Its average transmissivity is about 38.2%. Hence, the medium absorbs more energy from the probe beam than that of the case in Fig. 3(a1). It is due to the increased two-photon absorption in EIT system. The diffraction intensity corresponding to this absorption-phase grating is plotted in Fig. 4(a2). Comparing Fig. 4(a2) with Fig. 3(a2), we can see that more probe energy is deflected into the second order diffraction peaks. This is due to the improvement of phase modulation. When the detunings are δ₁ = 1.5γ^′ and δ₂ = −1.5γ^′, we can see that there is no amplitude modulation as shown in Fig. 4(b1) with the dashed line. Consequently, a pure-phase grating is built up in the medium. Almost all the energy in the probe beam is diffracted into the first-order and second-order directions with nearly no light left in zeroth-order [see Fig. 4(b2)]. The diffraction efficiency is enhanced by comparing with Fig. 3(a2). In Figs. 4(c1) and 4(c2), we also plot, respectively, the transmission function |T(x)| and diffraction pattern I(θ) for detunings δ₁ = 2.5γ^′ and δ₂ = −0.5γ^′. There is a periodical gain modulation for probe field, which leads to a structure of gain-phase grating, and therefore the diffraction efficiencies of high-order directions are significantly enhanced. As shown in Fig. 4(c2), the corresponding values of each high-order diffraction intensity increase to I(θ₁) = 450% at sin θ₁ = ±0.25 and I(θ₂) = 306% at sin θ₂ = ±0.5.

Fig. 4.

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Fig. 4. (color online) Amplitude |T(x)| (red dashed line) and phase ϕ(x) (blue solid line) of the transmission function T(x) plotted over two space periods for different two-photon detunings: (a1) δ1 = 0.5γ′, δ2 = −1.5γ′; (b1) δ1 = 1.5γ′, δ2 = −1.5γ′; (c1) δ1 = 2.5γ′, δ2 = −0.5γ′ when Ωs = 5.8γ. [(a2), (b2), and (c2)] Diffraction patterns I(θ) as a function of sin θ for the (a1), (b1), and (c1) cases. Other parameters are the same as those in Fig. 3.

Additionally, comparing the corresponding graphs in Figs. 3 and 4, it is found that the depth of phase modulation and the diffraction efficiency are dependent on the Rabi frequency of coupling field. The phase modulation makes the probe light mainly disperse into the high-order directions. Meanwhile, the amplification of probe beam from the Raman gain can enhance the diffraction efficiencies of all orders.

In order to further clearly clarify the effect of coupling field intensity on diffraction efficiency, we display the variations of the different order diffraction intensities I(θ₀), I(θ₁), and I(θ₂) with coupling field intensity Ω_s for different values of δ₁ and δ₂ in Fig. 5. It is found that the I(θ₀) decreases as Ω_s increases as shown with the solid lines in Figs. 5(a) and 5(b) for the absorption-phase gratings and phase gratings. But for the case of gain-phase grating (δ₁ = 2.5γ^′ and δ₂ = −0.5γ^′, the zeroth-order diffraction intensity I(θ₀) firstly presents two gain peak at a certain Ω_s, beyond which it declines for large Ω_s as shown with solid line in Fig. 5(c). Then the high-order diffraction intensities I(θ₁) and I(θ₂) can successively increase as Ω_s further increases, and they can take their maximum values under an appropriate Ω_s with the dashed and dashed-dotted lines shown in Figs. 5(a)–5(c), respectively. Moreover, the maximum values of I(θ₁) and I(θ₂) are not equivalent to each other for different conditions of δ₁ and δ₂. As a result, the probe light can be diffracted into first-order and higher order directions and the diffracting power of gratings can be also enhanced at the same time by modulating the intensity Ω_s of coupling fields. It is because the phase modulation is enlarged with the increase of Ω_s, which makes more probe energy deflect into the high-order directions. Therefore, these gratings can be applied to the all-optical multi-channel beam splitting and switching for optical networking and communication.

Fig. 5.

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	Fig. 5. (color online) Diffraction intensities of zeroth-order I(θ0) (black solid line), first-order I(θ1) (blue dashed line), and second-order I(θ2) (red dashed-dotted line) versus coupling field intensity Ωs for (a) δ1 = 0.5γ′, δ2 = −1.5γ′; (b) δ1 = 1.5γ′, δ2 = −1.5γ′; (c) δ1 = 2.5γ′, δ2 = −0.5γ′. Other parameters are the same as those in Fig. 3.

In this paper, by using two standing-wave fields, we propose a scheme for coherently induced grating in a mixture of two three-level atomic species with EIT and ARG systems. Three types of phase gratings are investigated in a real atomic system with ⁸⁵Rb and ⁸⁷Rb. By using the interaction between the two two-photon resonances, we obtain an enhanced refractive index in the medium. As a result of intensity dependent standing-wave fields, amplitude and phase are modulated periodically in space, and then a sinusoidal grating will be formed. It is shown that different diffraction patterns can be obtained by varying the two-photon detunings δ₁ and δ₂. The two-photon absorption in EIT system and Raman gain in ARG system contribute to the amplitude modulation, which affects the diffraction efficiency of the induced gratings. Meanwhile, we can achieve different phase-shift values by changing the intensities of two standing-wave fields. A larger phase modulation depth can help to deflect the probe energy into the higher order diffraction directions. The intensities of two standing-wave fields have an important influence on the diffraction efficiency of gratings, thereby the diffraction intensities I(θ₀), I(θ₁), and I(θ₂) of different orders can take their maximum values at appropriate Ω_s. Therefore, it is necessary to choose appropriate parameters to achieve the required diffraction and splitting of probe beam. Thus, the proposed gratings may be employed as all-optical splitter and switching in optical networking and communication. Moreover, in addition to cold atoms, EIG can be formed in thermal atoms.^[19,46] Hence, our scheme can also be performed with hot rubidium atoms.^[42]