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Modern Physics Letters B

SCATTERING AND TUNNELING IN ACTIVE SYSTEMS

Y. BENNABI, Amine B. Hammou, N. Zekri

Published Mar 31, 2011• 2 citations
Publisher

Abstract

The scattering properties of one-dimensional potential with gain are studied by using a Schrödinger-like equation. The corresponding Hamiltonian is non-Hermitian with a real energy spectrum. The amplification-absorption duality previously observed is interpreted in terms of the transmission and reflection phases. For a rectangular barrier, the transmission phase oscillates with the barrier width as for passive systems, but the oscillations period is significantly reduced in the absorption region. In this region the reflection phase vanishes and the multiple scattering and interferences dominate. The gain effect is also investigated for double barrier structures as well as superlattices with active potentials. It is found that resonant tunneling energy and the mini-band width are not influenced by the gain, but the transmission is enhanced for small values of the potential imaginary part. For large values, the resonant transmission significantly decreases and the mini-bands disappear.

DOI
10.1142/s0217984911026474

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The scattering properties of one-dimensional potential with gain are studied by using a Schrödinger-like equation. The corresponding Hamiltonian is non-Hermitian with a real energy spectrum. The amplification-absorption duality previously observed is interpreted in terms of the transmission and reflection phases. For a rectangular barrier, the transmission phase oscillates with the barrier width as for passive systems, but the oscillations period is significantly reduced in the absorption region. In this region the reflection phase vanishes and the multiple scattering and interferences dominate. The gain effect is also investigated for double barrier structures as well as superlattices with active potentials. It is found that resonant tunneling energy and the mini-band width are not influenced by the gain, but the transmission is enhanced for small values of the potential imaginary part. For large values, the resonant transmission significantly decreases and the mini-bands disappear.
We investigate the localization observed recently for locally non-hermitian Hamiltonians by studying the effect of the amplification on the scaling behavior of the transmission and reflection phases in 1D periodic chains of δδδ-potentials. The amplification here is represented by an imaginary term added to the on-site potential. It is found that both phases of the transmission and reflection amplitudes are strongly affected by the amplification term. In particular, the phases in the region of amplification become independent of the length scale while they oscillate strongly near the maximum transmission (or reflection). The interference effects on the phase in passive systems are used to interpret those observed in the presence of amplification. The phases of the transmission and reflection are found to oscillate in passive systems whith increasing periods in the allowed band for the transmission phase while for the reflection phase, its initial value is always less than π/2π/2π/2 in this band.
The Schrödinger equation is reformulated as a universal continuity equation, which connects between changes in the particles probability density distribution to probability current densities (fluxes). The formulation of particle conservation in terms of stationary fluxes enables one to associate stationary wave functions also to open quantum systems characterized by stationary particle currents. These functions are (improper) solutions of the stationary Schrödinger equation, obtained under scattering boundary conditions. These boundary conditions can be fulfilled for any positive asymptotic kinetic energy, hence, the energy spectrum of the scattering states is continuous. We demonstrate flux calculations in scattering through a one-dimensional potential energy well/barrier, focusing on transmission and reflection probabilities. Nonclassical phenomena such as transmission at energies below a potential energy barrier (quantum tunneling), or reflections at energies above a potential energy well are analyzed. The phenomenon of full transmission through a double barrier structure (resonant tunneling) is introduced in the context of nanoscale transport.
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We identify a non-Hermitian counterpart of dynamical tunnelling induced by a localised absorbing region in a symmetric confining potential. The absorbing region, described by an imaginary potential barrier, induces an effective double-well structure with a pair of long-lived states that have little population in the barrier. In contrast to the periodic tunnel oscillations through a real barrier, the system slowly evolves towards equal population of the two sides, while the absorbing region remains only weakly populated. We demonstrate this generic mechanism in two continuous potentials, and capture its essence analytically in a minimal 3×33\times 33×3 matrix model. We propose a realistic optical waveguide setup for the experimental observation of the effect.
Two rectangular models described by the one-dimensional Schr\"odinger equation with sharply localized potentials are suggested. The potentials have a multilayer thin structure being composed from adjacent barriers and wells. Their peculiar tunneling properties are studied in considerable detail. Particularly, in the zero-range limit when the potentials are squeezed to a single point, sharp peaks with total transmission are observed at certain (positive and negative) quantized values of the potential strength constant forming infinite discrete sets. Beyond these sets, the barrier-well structures behave as a perfectly reflecting wall. The transcendental equations with respect to potential strengths, the solutions of which determine transmission (resonance) sets, are derived. In this regard, both the models are exactly solvable. The energy dependence of an incident particle is shown to reveal a resonance behavior, being completely different from that observed in a typical double-barrier structure.
The comprehension of the tunnel effect, the passage of matter waves and particles across a high potential barrier, was the most astounding of all discoveries in quantum mechanics. Research on tunnelling in semiconductors and superconductors, as well as the development of scanning tunnelling microscopy, garnered five Nobel prizes in physics. In this study, the resonant tunnelling effect of electron tunnelling through a one-dimensional double-potential barrier was found and verified by solving the time-independent Schrödinger equation and calculating the transmittance rate of the electron tunnelling through the two potential barriers by using the transmission matrix technique performed by MATLAB. According to the analysis, the transmittance rate was one under certain specific conditions (under some specific incident energy values or some specific distance between two potential barriers), which was called as resonant tunnelling. However, in this study, square potential wells were analyzed, which can be considered merely as ideal models in real life, and therefore all results are idealized. For further investigations, researchers can use more realistic parameters to set up potential wells and analyze real-life problems.