A system with non-Hermitian Hamiltonian commutative with the parity-time operator, proposed by Bander et al., has a real eigenenergy spectrum and some novel properties under certain conditions. Due to the similarity between Schrodinger's equation and the optical Helmholtz equation, the optical system with out-of-phase spatial modulation is a good platform to simulate a system with parity-time (PT) symmetry, which is named the non-Hermitian optical system. In recent years, non-Hermitian optical structures based on discrete systems such as optical waveguide, hybrid optical micro-cavity, electrical circuit resonators, and continuous optical media such as cold atomic ensemble with spatially periodic modulation, have been implemented successively in experimental and theoretical studies. Spectroscopic devices such as diffraction grating have been a significant branch of optical devices since Newton's era. Electromagnetically induced grating (EIG) based on electromagnetically induced transparency (EIT) makes it possible to tune the diffraction patterns dynamically. In recent years, combined with the non-Hermitian optical modulation, many schemes of one- or two-dimensional asymmetric optical diffraction gratings have been proposed successively. However, due to rigid realization conditions of PT symmetry or PT antisymmetry, there is still a great hindrance to realizing precise and flexible dynamic operation, especially for some special optical diffractions. In most previous schemes, dual spatial periodic modulation (via amplitude, detuning of coupling field, or atomic density) has been adopted to achieve two goals, including the realization of PT symmetry or PT antisymmetry and the construction of a grating structure. This results in the lack of accurate modulation capabilities with the protection of PT symmetry or PT antisymmetry. Therefore, a method for preparing non-Hermitian EIG with simple structures easy to analyze, dynamic control ability, and protection of optical non-Hermitian symmetry is necessary and desired.

We consider an ensemble of cold ⁸⁷Rb atoms driven into a three-level Lambda configuration by two coherent fields with frequencies ωp and ωc. The weak probe field ωp interacts with transition g?e, while the strong control field ωc acts upon transition m?e. The states g and e are coupled by the incoherent pumping Λicp additionally. By periodically modulating the coupling field detuning Δc(x), it is possible to get an asymmetric diffraction pattern in this system.

According to the study on far-field diffraction characteristics of probe field through the system, some results are as follows. 1) When the parameters are chosen as Δ_p=-2.27×2π MHz, Δ_c0=-0.1×2π MHz,δ_c0=1.0×2π MHz and ψ=0, we can get the PT symmetric susceptibility. Then the modulations for the real part and imaginary part of the susceptibility are out-of-phase, and the lopsided diffraction patterns are shown in Fig. 2(b). 2) When the parameters are chosen as Δ_p=Δ_c0=0, δ_c0=1.0×2π MHz, and ψ=0, we can get the PT antisymmetric susceptibility. The modulations for the real part and imaginary part of the susceptibility are still out-of-phase, resulting in the lopsided diffraction patterns shown in Figs. 3(b₁) and 3(b₂). 3) With increasing optical depth, the diffracted intensity is transferred from zero order to one order, while with increasing incoherent pumping, the diffracted intensity is transferred from one order to zero order in the PT antisymmetric grating as shown in Fig. 4. 4) The initial phase of coupling field detuning can modulate the diffraction in two ways. The first one is changing the diffraction direction from a negative angle to a positive angle by varying the initial phase of the coupling field detuning from 0 to π. The other one is tuning the diffraction symmetries of the grating. When ψ=mπ, we can get asymmetric diffraction patterns, and when ψ=(2m-1)π/2, we can get symmetric diffraction patterns shown in Fig. 5.

We propose a theoretical scheme of non-Hermitian electromagnetically induced grating based on incoherent pumping. The system consists of an ultra-cold atomic ensemble with the Lambda-type three-level structure and an incoherent pumping field. Combined with incoherent pumping, non-Hermitian symmetries of the system including optical PT symmetry and PT antisymmetry under the single spatial period modulation can be implemented. During the research on far-field diffraction characteristics of probe field through the system, we can draw the following conclusions. First, a switch between different non-Hermitian optical symmetries can be attained by controlling the detuning of the coupling field. Secondly, the diffraction efficiency is effectively modulated by incoherent pumping at a constant optical depth, which introduces a new degree of freedom of grating manipulation. In addition, tuning the initial phase of coupling field detuning can effectively modulate the diffraction symmetry and diffraction pattern of the system. The theoretical results not only can facilitate the research and development of non-Hermitian optics and scattering-type all-optical devices but also can be applied in quantum optics and quantum information processing.