Enhanced cross-Kerr nonlinearity induced $PT$-symmetry in optical lattices

We propose a physical scheme to realize optical lattices that are symmetric under a combination of parity and time ($PT$) reversal. The $PT$-symmetric structures can be constructed with periodical gain and loss profiles in a coherently prepared four-level N-type atomic system under the effects of cross-Kerr nonlinearity and Doppler broadening. The atoms are considered in the Gaussian density distribution loaded into the dipole traps of one- and two-dimensional optical lattices. The spatial modulation of the probe field susceptibility is achieved by a proper combination of standing wave strong control fields. With proper detunings and coupling field Rabi frequencies, we find the conditions of achieving $PT$-symmetry in optical lattices of nonlinear atoms. We show that the $PT$-symmetry can be realized when the cross-Kerr nonlinearity is significantly enhanced, and further demonstrate that it survives the Doppler broadening. Our model is based on experimentally observable and effectively controllable scheme which may find potential applications in optical information processing. (paper)