Parity-time symmetry with coherent atomic gases: a concise review on recent progress

In standard quantum theory, the Hamiltonian describing a physical system is assumed to be Hermitian in order to guarantee the energy spectrum to be real and the time evolution to be unitary. In recent years, it was recognized that non-Hermitian Hamiltonians with parity-time () symmetry can exhibit entirely real spectra, raising the possibility for extending the quantum theory to complex domain and hence stimulated growing interest in recent years. Many proposals have been presented for realizing -symmetric Hamiltonians in various physical systems. Among them -symmetric coherent atomic gases are special and possess many unique advantages, including the possibility to obtain authentic -symmetric refractive indexes (which have balanced gain and loss in the whole space), the capability to actively control and precisely manipulate system parameters in situ, and the feasibility to acquire large Kerr nonlinearity based on the resonance character between light and atoms. In this article, we review various schemes for the realization of symmetry with coherent atomic gases, elucidate their interesting properties and promising applications. In particular, the non-linear optical effect in the -symmetric atomic gases are described, which may be served as useful building blocks for developing novel photonic devices with active light control at very low power level.

Two dimension PT symmetry spacial soliton in atomic gases with linear and nonlinear potentials

We propose a scheme to realize a parity-time (PT) symmetric nonlinear system in a coherent atomic gas via electromagnetically induced transparency. We show that it is possible to construct an optical potential with PT symmetry due to the interplay among the Kerr nonlinearity stemmed from the atom-photon interaction, the linear potential induced by a far-detuned Stark laser field, and the optical gain originated from an incoherent pumping. Since the real part of the PT-symmetric potential depends only on the intensity of the probe field, the potential is nonlinear and its PT-symmetric properties are determined by the input laser intensity of the probe field. Moreover, we obtain the fundamental soliton solutions of the system and attain their stability region in the system parameter space. The dependence of the exceptional point (EP) location on the soliton maximum amplitude is also illustrated. The research results reported here open a new avenue for understanding the unique properties of PT symmetry of a nonlinear system. They are also promising for designing novel optical devices applicable in optical information processing and transmission.

We studied the specular and nonspecular remittances and the absorptance of two-dimensional photonic crystals (PCs) comprising a periodic array of dielectric rods immersed in a coherent atomic gas (CAG). Illumination by obliquely incident, s-polarized plane waves was considered. Minipassbands that may include peaks of nearly complete transmission appear within the stop bands of the corresponding gas-free PCs, due to the strong frequency dispersion in the relative permittivity of the CAG. As a result, high-efficiency passbands of a gas-free PC and those arising due to the CAG can coexist. The latter can be tuned in a simpler way than by varying the constitutive parameters of the CAG, i.e., via variation of the incidence angle. In contrast with the earlier studied metallic PCs immersed in a CAG, new passbands may also appear when the CAG behaves as a medium with ultralow positive permittivity. Also, complete absorptance bands (with zero overall reflectance and overall transmittance) can be obtained, which are well correlated with the frequency-dependent characteristics of the CAG.

We propose a realistic physical scheme to realize linear Gaussian optical potential with parity-time (PT) symmetry and two dimensional (2D) spacial solitons in a coherent atomic gas. It is shown that the PT-symmetric potential can be created through the spatial modulation of the control and relevant atomic parameters. We find that the Gaussian PT potential parameters, the imaginary part and the width and the position, play crucial roles in the occurrence of the PT phase transition. We demonstrate that the system supports stable 2D dipole solitons and vortex solitons, which can be managed via tuning PT potential. Furthermore, the dynamic characteristics of the symmetric scatter and collision of solitons are shown.

Understanding and manipulating the non-Hermitian optical property based on coherent atomic gases is of great importance and has attracted much theoretical and experimental attentions. Advancing this study to the nonlinear optics regime is highly desirable due to its importance in fundamental physics and potential applications. In this work, we propose to realize a tunable electromagnetically induced grating (EIG) with parity-time ($\mathcal{PT}$) symmetry in a cold gas of Rydberg atoms, where interatomic interactions between Rydberg states are mapped to strong and long-range optical interactions, and investigate nonlinear light diffractions in this system. We show that for far-field diffraction, laser beams incident upon the $\mathcal{PT}$-symmetric EIG display distinctive asymmetric diffraction fringes, which can be actively manipulated through tuning the gain-absorption coefficient of the EIG, the incident intensity of the laser beam, and the nonlocality provided by Rydberg atoms. For near-field diffraction, the nonlinear Talbot diffraction carpets emerge and can be modulated by $\mathcal{PT}$ symmetry in the presence of strong nonlocal interactions, allowing the realization of controllable optical self-imaging. The results are not only imperative for the study of non-Hermitian nonlinear optics but also useful for characterizing the interatomic interaction in Rydberg gases and for designing new optical devices useful in optical information processing and transmission.

The electromagnetic transmission, reflection, and absorption characteristics of two-dimensional metallic photonic crystals with a coherent atomic gas as the host medium were systematically studied, with emphasis on the appearance and features of mini passbands within bandgaps of the unfilled (gas-free) crystals. Only normally incident s-polarized plane waves were considered. The mini passbands are connected with strong frequency dispersion of the relative permittivity of the host gas, being highly variable for a certain narrow regime of frequencies. Transmission effects similar to those connected with defect modes can appear in photonic crystals, which are associated with localization of dispersion in the frequency domain rather than with spatial localization of the field at structural defects. In addition, analogy with Fabry–Perot resonances is possible within the new bands. Their locations can be strongly sensitive to a variation of gas parameters so that they can be efficiently tuned at fixed frequency. Also, the occurrence of high-absorbance bands can be correlated with the frequency-dependent properties of the metal and the coherent atomic gas. Finally, the energy of the incident wave can be distributed in a desired proportion between either transmittance or reflectance on one hand and absorbance on the other.

We propose a physical scheme to realize combined linear and nonlinear optical potentials with parity-time ($\mathcal{PT}$) symmetry and investigate the scattering property of optical solitons in a coherent atomic gas. We show that the combined linear and nonlinear $\mathcal{PT}$-symmetric potentials can be created through the spatial modulation of the control laser field and the inclusion of the Kerr nonlinearity of the signal laser field. We demonstrate that the imaginary part of the nonlinear $\mathcal{PT}$ potential plays a crucial role for the occurrence of the $\mathcal{PT}$ phase transition and the change of the $\mathcal{PT}$ phase diagram, which can be actively manipulated in our system. We demonstrate also that the system supports stable optical solitons, which can be managed via tuning the combined linear and nonlinear $\mathcal{PT}$ potentials; furthermore, by taking the combined linear and nonlinear $\mathcal{PT}$ potentials as a defect, the scattering of the optical solitons by the defect displays evident asymmetric behavior, controlled by the imaginary parts of the combined linear and nonlinear $\mathcal{PT}$ potentials. The results reported here may have potential applications in optical information processing and transmission.

We propose a method for trapping weak signal pulses by soliton and realizing its trajectory control via electromagnetically induced transparency (EIT). The system we consider is a cold, coherent atomic gas with a tripod or multipod level configuration. We show that, due to the giant enhancement of Kerr nonlinearity contributed by EIT, several weak signal pulses can be effectively trapped by a soliton and cotravel stably with ultraslow propagating velocity. Furthermore, we demonstrate that the trajectories of the soliton and the trapped signal pulses can be manipulated by using a Stern-Gerlach gradient magnetic field. As a result, the soliton and the trapped signal pulses display a Stern-Gerlach deflection and both of them can bypass an obstacle together. The results predicted here may be used to design all-optical switching at very low light level.

A robust light storage and retrieval (LSR) in high dimensions is highly desirable for light and quantum information processing. However, most schemes on LSR realized up to now encounter problems due to not only dissipation, but also dispersion and diffraction, which make LSR with a very low fidelity. Here we propose a scheme to achieve a robust storage and retrieval of weak nonlinear high-dimensional light pulses in a coherent atomic gas via electromagnetically induced transparency. We show that it is available to produce stable (3 + 1)-dimensional light bullets and vortices, which have very attractive physical property and are suitable to obtain a robust LSR in high dimensions.

Wideband switchable diode-like transmission can be exhibited by an asymmetric dielectric photonic crystal, when the host medium is changed from air to a coherent atomic gas (CAG), a strongly dispersive medium. Significant modification of diffraction-enabled one-way transmission due to the CAG is possible in both frequency and incidence-angle domains in the short-wave infrared regime. In particular, new one-way and high-contrast passbands, which are as much as 1.0 THz in bandwidth, can appear at fixed incidence angle within a stop band of the CAG-free structure and tuned by varying the oscillator strength of the CAG. These passbands correspond to relatively small, either positive or negative, values of the dielectric susceptibility of the CAG.

We propose a scheme to realize parity-time ($\mathcal{PT}$) symmetry and nonlocal optical solitons in a cold Rydberg atomic system with electromagnetically induced transparency. We show that a two-dimensional (2D) periodic optical potential with $\mathcal{PT}$ symmetry can be obtained for the propagation of probe laser field by using an incoherent population pumping between two low-lying levels and spatial modulations of control and assistant laser fields. We also show that, based on the giant nonlocal Kerr nonlinearity originated from the strong, long-range atom-atom interaction, the system supports 2D nonlocal gap solitons with very low light intensity. In particular, we find that the degree of the nonlocality of the Kerr nonlinearity, which can be actively tuned in our system, can be used to manipulate the phase transition of the $\mathcal{PT}$ symmetry and the behavior of the nonlocal optical solitons. Our study opens a route for developing non-Hermitian nonlinear optics, especially for realizing and controlling high-dimensional weak-light optical solitons through adjustable $\mathcal{PT}$ symmetry and giant nonlocal optical nonlinearity.

Electromagnetically induced transparency (EIT), a typical quantum interference effect, has been extensively investigated in coherent atomic gases. In recent years, it has been recognized that the plasmonic analog of atomic EIT, called plasmon-induced transparency (PIT), is a fruitful platform for the study of EIT-like propagation and interaction of plasmonic polaritons. Many proposals have been presented for realizing PIT in various metamaterials, which possess many unique characters, including the suppression of absorption of electromagnetic radiation, the reduction of propagation velocity, etc. Especially, nonlinear PIT metamaterials, obtained usually by embedding nonlinear elements into meta-atoms, can be used to acquire an enhanced Kerr effect resulted from the resonant coupling between radiation and the meta-atoms and to actively manipulate structural and dynamical properties of plasmonic metamaterials. In this article, we review recent research progress in nonlinear PIT metamaterials, and elucidate their interesting properties and promising applications. In particular, we give a detailed description on the propagation and interaction of nonlinear plasmonic polaritons in metamaterials via PIT, which are promising for chip-scale applications in information processing and transmission.

We theoretically investigate one-dimensional localized gap modes in a coherent atomic gas where an optical lattice is formed by a pair of counterpropagating far-detuned Stark laser fields. The atomic ensembles under study emerge as Λ-type three-level configuration accompanying the effect of electromagnetically induced transparency (EIT). Based on Maxwell-Bloch equations and the multiple scales method, we derive a nonlinear equation governing the spatial-temporal evolution of the probe-field envelope. We then uncover the formation and properties of optical localized gap modes of two kinds, such as the fundamental gap solitons and dipole gap modes. Furthermore, we confirm the (in)stability regions of both localized gap modes in the respective band-gap spectrum with systematic numerical simulations relying on linear-stability analysis and direct perturbed propagation. The predicted results may enrich the nonlinear horizon to the realm of coherent atomic gases and open up a new door for optical communication and information processing.

We propose a scheme to realize the storage and retrieval of optical Peregrine solitons in a coherent atomic gas via electromagnetically induced transparency (EIT). We show that optical Peregrine solitons with very small propagation loss, ultraslow motional velocity, and extremely low generation power can be created in the system via EIT. We also show that such solitons can be stored, retrieved, split, and routed with high efficiency and fidelity through the manipulation of control laser fields. The results reported here are useful for the active control of optical Peregrine solitons and promising for applications in optical information processing and transmission.