We show that a vapor of multilevel atoms driven by far-off resonant laser beams, with possibility of interference of two Raman resonances, is highly efficient for creating parity-time (PT ) symmetric profiles of the probe-field refractive index, whose real part is symmetric and imaginary part is antisymmetric in space. The spatial modulation of the susceptibility is achieved by proper combination of standing-wave strong control fields and of Stark shifts induced by a far-off-resonance laser field. As particular examples we explore a mixture of isotopes of Rubidium atoms and design a PT -symmetric lattice and a parabolic refractive index with a linear imaginary part.PACS numbers: 42.50. Gy, 42.65.An, 11.30.Er While non-Hermitian operators obeying pure real spectra, like for example the Bogoliubov-de Gennes [1] spectral problem, linear stability problem for nonlinear waves, or simply a parabolic potential with linear imaginary part [2], are known in physics for long time, it was only due to the work [3] that fundamental importance of such operators became widely recognized. It was discovered in [3] that there exists a wide class of complex potentials of Schrödinger equation obeying pure real spectra, and even most importantly, that this property is intrinsically related to the parity (P) and time (T ) symmetries of physical systems. This discovery triggered the discussion [4] on the fundamentals of quantum mechanics whose axioms are based on Hermitian operators for observables. Further growth of interest in the theory of parity-time (PT ) symmetric potentials was originated by suggestions of implementation of PT symmetry in a waveguide with gain and absorption [5], which was based on the analogy between quantum mechanics and paraxial optics where the refractive index plays the role of the potential in the Schrödinger equation. In optics PT -symmetric refractive index profiles (i.e. obeying gain and losses of a given geometry) has been experimentally realized using process of four-wave mixing in Fe-doped LiNbO 3 substrate [6]. The possibility of optical realization of PT symmetric potentials motivated various suggestions of practical applications, like nonreciprocal wave propagation [6,7], implementation of coherent perfect absorber [8], giant wave amplification [9], etc. Experimental realization of PTsymmetry using plasmonics [10] and temporal simulation of lattices using optical couplers [11] were also reported.In this Letter we demonstrate the possibility of practical implementation of spatially distributed PTsymmetric refractive index, i.e. the one having the property n(x) = n * (−x), in vapors of multi-level atoms driven by control fields with properly chosen Raman resonances and by a far-off-resonant Stark field.First we recall some recent achievements in creation of large susceptibilities in atomic vapors controlled by external laser beams. While such systems are intrinsically dissipative, it was suggested in [12] and shown experimentally in [13], that using the destructive interference in imaginary part ...
We present a systematic study on the dynamics of a ultraslow optical soliton in a cold, highly resonant three-state atomic system under Raman excitation. Using a method of multiple scales we derive a modified nonlinear Schrödinger equation with high-order corrections that describe effects of linear and differential absorption, nonlinear dispersion, delay response of nonlinear refractive index, diffraction, and third-order dispersion. Taking these effects as perturbations we investigate in detail the evolution of the ultraslow optical soliton using a standard soliton perturbation theory. We show that due to these high-order corrections the ultraslow optical soliton undergoes deformation, change of propagating velocity, and shift of oscillating frequency. In addition, a small radiation superposed by dispersive waves is also generated from the soliton. The results of the present work may provide a guidance that is useful for experimental demonstration of ultraslow optical soliton in cold atomic systems.
We develop a systematic analytical approach to consider the dynamics of linear and nonlinear excitations in trapped quasi-one-dimensional Bose-Einstein condensates with repulsive atom-atom interactions. We show that, for a condensate strongly confined in two transverse directions, the ground state of the system involves the high-order eigenmodes of the transverse confining potential in the transverse directions and effective highorder Thomas-Fermi wave functions in the axial direction. The linear excitations of the system have a Bogoliubov-type spectrum with the excitation frequency varying slowly along the axial direction. We find that, in a weak nonlinear approximation, the amplitude of a nonlinear excitation is governed by a variable coefficient Korteweg-de Vries equation with additional terms contributed from the transverse structure and the inhomogeneity in the axial direction of the condensate, which results in varying amplitude, width, and velocity for dark solitons. Because of the inhomogeneity the dark solitons undergo deformation and emit radiations when traveling along the axial direction. We finally demonstrate that a dark soliton will disintegrate into several ones plus a residual wave train when passing over a steplike potential.
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