May 2012
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Day May 16, 2012

Complex Trajectories in a Classical Periodic Potential

Alexander G. Anderson, Carl M. Bender

This paper examines the complex trajectories of a classical particle in the potential \(V(x)=-\cos(x)\). Almost all the trajectories describe a particle that hops from one well to another in an erratic fashion. However, it is shown analytically that there are two special classes of trajectories \(x(t)\) determined only by the energy of the particle and not by the initial position of the particle. The first class consists of periodic trajectories; that is, trajectories that return to their initial position \(x(0)\) after some real time \(T\). The second class consists of trajectories for which there exists a real time \(T\) such that \(x(t+T)=x(t) \pm2 \pi\). These two classes of classical trajectories are analogous to valence and conduction bands in quantum mechanics, where the quantum particle either remains localized or else tunnels resonantly (conducts) through a crystal lattice. These two special types of trajectories are associated with sets of energies of measure 0. For other energies, it is shown that for long times the average velocity of the particle becomes a fractal-like function of energy.
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Light Transport in Random Media with PT-Symmetry

Samuel Kalish, Zin Lin, Tsampikos Kottos

The scattering properties of randomly layered optical media with \({\cal PT}\)-symmetric index of refraction are studied using the transfer-matrix method. We find that the transmitance decays exponentially as a function of the system size, with an enhanced rate \(\xi_{\gamma}(W)^{-1}=\xi_0(W)^{-1}+\xi_{\gamma} (0)^{-1}\), where \(\xi_0(W)\) is the localization length of the equivalent passive random medium and \(\xi_{\gamma}(0)\) is the attenuation/amplification length of the corresponding perfect system with a \({\cal PT}\)-symmetric refraction index profile. While transmitance processes are reciprocal to left and right incident waves, the reflectance is enhanced from one side and is inversely suppressed from the other, thus allowing such \({\cal PT}\)-symmetric random media to act as unidirectional coherent absorbers.
Optics (physics.optics)