The application of interference and atomic coherence have led to innovations in the fields of lasers and material science, resulting in the development of noise-free quantum lasers [1, 2] and novel optical materials such as phaseonium, which exhibits a significantly increased index of refraction [3]. These fundamental breakthroughs offer the potential to fabricate novel devices that rely on quantum coherence and interference. Quantum coherence and interference also play a crucial role in modifying optical phenomena in quantum systems. One such phenomenon is electromagnetically induced transparency (EIT), where absorption of a weak probe field in a medium is eliminated due to coherence induced by a strong field [4–6]. EIT in an atomic system is utilized as an optical memory in nonlinear optics [7, 8] and quantum information processing [9], typically using a three-level Λ-type atomic system. However, a pulse traveling through this system experiences absorption, rendering it unsuitable for practical applications. To overcome this issue, a microwave field can be applied in the lower two levels, which not only compensates for absorption but also leads to amplification in EIT [10].

An atomic medium can act as a diffraction grating through the phenomenon of electromagnetically induced grating (EIG), which occurs when a standing-wave field is used instead of a control driving field in EIT configuration [11]. The spatial distribution of the standing-wave field creates periodic fluctuations in the probe field's magnitude, resulting in the diffraction of an optical field into higher orders. The EIG phenomenon has been demonstrated experimentally in an atomic medium composed of sodium atoms [12]. Consequently, there is growing interest in exploring EIG and its potential applications, including the creation of a free background [12], electromagnetically induced phase grating in a three-level [13] and a four-level double dark resonance system [14], and employing EIG in all-optical two-port signal switch [15–17]. In recent decades, numerous atomic configurations have been suggested for investigating EIG [13, 14, 18–27].

Here, we propose a new scheme to explore EIG in a coherently prepared atomic system called phaseonium [28, 29]. A natural question arises concerning the benefits that our proposed scheme offers over previously introduced models. Our proposed scheme employs a three-level atomic model, where the atoms are initially set up in a state of coherent superposition of the lower energy levels. As a result, absorption can be switched into gain (i.e. amplification) by controlling the relative phase of the two lower levels. This phase-dependent coherent atomic ensemble represents a new state of matter that has direct applications, including lasing without population inversion and quantum noise reduction. These ideas hold promise for high-precision magnetometry and particle acceleration, both of which are areas of particular interest [29].

In this manuscript, we analyze the diffraction of light in an ensemble of Λ-type atomic configuration, also known as the phaseonium medium. Due to the coherent superposition of lower levels, the atoms form a closed loop, with the relative phase playing a vital role in manipulating the optical characteristics of the medium. According to our observations, the intensity of the first order diffraction is increased when the relative phase is varied between 0 and π. In addition, the first-order diffraction may be amplified by increasing its interaction length as well as the intensity of the optical field generated by the medium.

The proposed scheme for the realization of EIG is depicted in figure 1. A medium comprising a Λ-type atomic configuration with three energy levels, labeled as $\left|1\right\rangle $, $\left|2\right\rangle $, and $\left|3\right\rangle $, interacts with two optical fields, E₁ and E₂(x) (see figure 1(a)). The atomic transition $\left|1\right\rangle \leftrightarrow \left|3\right\rangle $ is driven by the optical field E₁ having Rabi frequency Ω₁ = E₁μ₁₃/2ℏ, where μ₁₃ represents the corresponding dipole matrix element. The other transition $\left|2\right\rangle \leftrightarrow \left|3\right\rangle $ is coupled by the position-dependent optical field E₂(x) with Rabi frequency ${{\rm{\Omega }}}_{2}(x)={{\rm{\Omega }}}_{2}\sin ({k}_{2}x)\,=\left({E}_{2}{\mu }_{23}/2{\hslash }\right)\left(\sin ({k}_{2}x)\right)$, where k₂ = 2π/λ₂ is the wave vector of the corresponding field. Our analysis is based on the initial assumption that the atoms are in a superposition of the lower levels, $\left|1\right\rangle $ and $\left|2\right\rangle $, at the start of the interaction. The spatial modulation of the standing-wave field Ω₂(x) induces periodic fluctuations in the intensity of the incident probe field, resulting in diffraction of the probe field as illustrated in figure 1(b).

The atom field interaction Hamiltonian for such a system in the appropriate rotating frame can be expressed aswhere δ₁ = ω₃₁ − ω₁ and δ₂ = ω₃₂ − ω₂ are the detunings of the first and second field respectively. Here ω₃₁ and ω₃₂ are the frequencies of the transitions from $\left|1\right\rangle $ to $\left|3\right\rangle $ and $\left|2\right\rangle $ to $\left|3\right\rangle $, respectively, and ω₁ and ω₂ are the frequencies of the first and second field. To describe the dynamics of Ω₁ and Ω₂, we need to calculate atomic coherences ρ₁₃ and ρ₂₃. Therefore, required rate equations for the proposed atomic model can be written asHere the symbols γ₃₁ and γ₃₂ represent the decay rates that correspond to the transition of an excited state denoted by $\left|3\right\rangle $ to lower energy states identified by $\left|1\right\rangle $ and $\left|2\right\rangle $, respectively. The atoms are presumed to be in an initial state that is a superposition of the two lower energy levels, which can be expressed aswhere φ₁ and φ₂ are the phase associated with the lower levels ∣1⟩ and ∣2⟩. To obtain the steady-state solution for the atomic coherence ρ₁₃, which is required for calculating the transmission of Ω₁, we use the definition of the density operator $\rho =\left|\psi \left(0\right)\right\rangle \left\langle \psi \left(0\right)\right|$. This gives ρ₃₃ ≈ 0, ρ₁₁ ≈ ∣C₁∣², ρ₂₂ ≈ ∣C₂∣² and ${\rho }_{12}\approx {C}_{1}{C}_{2}^{* }{{\rm{e}}}^{{\rm{i}}({\phi }_{2}-{\phi }_{1})}$, leading to the following expression for ρ₁₃The dielectric susceptibility $\chi =\tfrac{{\rm{N}}{\left|{\wp }_{13}\right|}^{2}}{{\epsilon }_{0}{\hslash }{{\rm{\Omega }}}_{1}}{\rho }_{13}$ of the proposed system by using equation (4) takes the formHere $\xi =\tfrac{{\rm{N}}{\left|{\wp }_{13}\right|}^{2}}{{\epsilon }_{0}{\hslash }}$ and φ = φ₁ − φ₂ represents the relative phase between the superposition of the lower states. The dielectric susceptibility χ (equation (5)) depends on standing wave field Ω₂(x), and the spatial modulation of susceptibility leads to diffraction of the Ω₁. The solution of Maxwell's equations under slowly varying envelope approximation for the propagation of E₁ in the z-direction, yields the diffraction pattern of E₁where λ₁ is the wavelength of E₁ and P₁ = ε₀χE₁ is the polarization induces by E₁. After some mathematical calculations, equation (6) reduces to [11, 23]where η(x)(ζ(x)) is the absorption (dispersion) coefficients of E₁ and defined as $\eta (x)=[\tfrac{\pi }{{\lambda }_{1}}]\mathrm{Im}[\chi ]$ and $\zeta (x)=[\tfrac{\pi }{{\lambda }_{1}}]\mathrm{Re}[\chi ]$ respectively. We can obtain the analytical expression for the transmission E(x) of E₁ by solving equation (7) for the interaction length z = L (which is equivalent to the thickness of the medium), and is given bywhere e^−η(x)L represents absorption and e^iζ(x)L represents phase modulation, respectively. Intensity distribution due to diffraction may be expressed as [11, 23]Here, N denotes the spatial periods count of the grating, θ signifies the diffraction angle, Λ_x denotes the spatial period along the x-axis, which can be represented as $[\tfrac{8\pi }{{k}_{x}}]$, and E(θ) represents the transform of E(x) using the Fourier method. One possible formulation to describe Fraunhofer diffraction in a single period is as follows:In order to study the intensity of the nth order diffraction, where $n=\tfrac{{{\rm{\Lambda }}}_{x}\sin (\theta )}{{\lambda }_{1}}$, we make use of I(θ_n), which represents the strength associated with the nth ordering diffraction angles and is represented withwithIn the following, we discuss the numerical results of EIG of our proposed model.

The objective of this section is to investigate the impact of the Rabi frequency strength and phase of an optical field on diffraction orders and to explore the realization of EIG in phaseonium. Firstly, we analyze medium's response to variations in the relative phase between two lower ground levels. Figure 2 illustrates the plot of the susceptibility's imaginary component with respect to the relative phase φ, which indicates the absorptive ($\mathrm{Im}[\chi ]\gt 0$) or gain ($\mathrm{Im}[\chi ]\lt 0$) nature of the medium. The remaining parameters are fixed at γ = 1 MHz, γ₃₁ = 2γ, δ₁ = 0, Ω₁ = Ω₂ = Ω = 1.8γ, ${C}_{1}=\tfrac{1}{\sqrt{2}}$, and ${C}_{2}=\tfrac{1}{\sqrt{2}}$. Our observations indicate that the proposed system undergoes a shift from absorptive to gain state with the increase in relative phase from 0 to π. Therefore, by modifying the relative phase, we may control the model's characteristics.

Next, we explore the achievement of EIG in an ensemble of three-level phaseonium atoms by investigating diffraction intensity distribution using equation (9). We begin with examining the diffraction pattern in a phase-shifted atomic medium. The diffraction intensity versus $\sin (\theta )$ is shown in figure 3 for four distinct values of relative phase φ: (a) φ = 0, (b) φ = π/3, (c) φ = π/2, and (d) φ = π. According to the figure, the angles at which zeroth as well as the first-order diffraction take place are θ = 0, 0.25 and −0.25 rad. It is worth noting that the magnitude of the zeroth order diffraction is considerably higher than that of the first order diffraction. The diffraction intensity of the zeroth and first order increases as the relative phase between the superposition of the ground level changes from 0 to π, as depicted in figure 3. In our proposed phaseonium model, at phase φ = 0, the medium exhibits an absorptive nature resulting in low transmission. However, at phase φ = π, the absorption converts in to gain, leading to an increase in transmission and diffraction intensity of the zeroth and first orders. We conducted additional analysis to examine the impact of relative phase on the intensity of first order diffraction. The findings are presented in figure 4, where the first order diffraction intensity is plotted against φ. For plotting this result, we set $\sin \theta =0.25$, while maintaining all other parameters as specified in figure 3. First order diffraction intensities are shown to grow between 0 to π, confirming our previous findings. The change in the medium's behavior from absorptive to gain is responsible for the increase in the first order diffraction intensity when the relative phase shifts in zero to π.

Our earlier results are validated by the observed increase in the first order diffraction intensity as the relative phase increases from 0 to π. The enhancement in the first order diffraction intensity by changing the relative phase from 0 to π is attributed to the shift in the medium's behavior from absorptive to gain.

In the preceding analysis, we have observed that the first order intensity reaches its maximum value at φ = π. Next, we investigate the effect of the Rabi frequency strength Ω₂ on the EIG diffraction intensity. Figure 5 displays diffraction intensity distribution for four distinct values of Ω₂: Ω₂ = 1γ, 1.5γ, 2γ, and 2.5γ against $\sin (\theta )$. All other parameters are same as given in figure 3. The plot demonstrates a positive correlation between the increase in Ω₂ and the amplification of both the central zeroth order intensity and the first order diffraction. Further, in figure 6, we plot first order diffraction against Ω, demonstrating that the first order diffraction increases with increasing Ω₂ strength. The underlying reason for this phenomenon is that Ω₂ enhances the gain in the system at the phase φ = π, leading to a boost in the transmission spectrum of the probe light beam and an increase in the first order diffraction intensity. Finally, figure 7 presents a 3D plot and corresponding density plot of diffraction intensity against $\sin \theta $ and Ω₂, showing an enhancement in the diffraction intensity of the zeroth and first orders with an increase in Ω₂.

Here we would like to mention that interaction length of the atomic medium also plays a significant role in the amplification of the zeroth as well as the first order diffraction intensities. This effect can be observed from the 3D and density plots presented in figures 8(a) and (b), respectively. The plots show that at small values of L, only the zeroth order diffraction is observed. However, by increasing L, the first order diffraction on both sides also becomes prominent, as depicted in figure 8.

In conclusion, we have investigated the effect of coherent superposition of lower level states on the transmission function and diffraction intensity of a light beam in a three level atomic system, referred to as phaseonium. The results showed that the relative phase of the lower level superposition states influenced the diffraction intensity of the zeroth and first orders, with maximum intensity observed at a relative phase of π. This behavior is due to the gain in the system resulting from the coherent superposition of lower levels. Furthermore, we have examined the effect of Rabi frequency Ω₂ and interaction length L on the diffraction intensity of EIG. Increasing the strength of field Ω₂ and interaction length L amplified the central zeroth order intensity as well as the first order diffraction. The proposed atomic grating in phaseonium has potential applications in providing speckle noise-free, dark current-free, and background-free diffraction.