| Summary: | 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 ($ \mathcal PT $) 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 $ \mathcal PT $-symmetric Hamiltonians in various physical systems. Among them $ \mathcal PT $-symmetric coherent atomic gases are special and possess many unique advantages, including the possibility to obtain authentic $ \mathcal PT $-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 $ \mathcal PT $ symmetry with coherent atomic gases, elucidate their interesting properties and promising applications. In particular, the non-linear optical effect in the $ \mathcal PT $-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.
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