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Acta Physica Polonica A

Entropy Associated with Local Stabilization of the Pulse Area in Multilevel Atomic Media

Abu Mohamed Alhasan

Published May 1, 2018• 2 citations
Publisher

Abstract

This paper provides an entropy analysis to resonant and short pulses propagation in four-level atomic medium.As an example, we take D1 transition in rubidium 87 Rb atoms including hyperfine structure.We show how to construct the time dependent Bloch-metric for each optical transition in the Liouville space.Furthermore, we attempt to relate local stabilization of the pulse area to the distribution of the space-entropy.

DOI
10.12693/aphyspola.133.1205

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This paper provides an entropy analysis to resonant and short pulses propagation in four-level atomic medium.As an example, we take D1 transition in rubidium 87 Rb atoms including hyperfine structure.We show how to construct the time dependent Bloch-metric for each optical transition in the Liouville space.Furthermore, we attempt to relate local stabilization of the pulse area to the distribution of the space-entropy.
We investigate short pulses propagation through multilevel atomic vapors. The medium is composed of an ensemble of four atomic levels generated by the hyperfine structure of the D1 transition in Rubidium-87 atoms. The pulses are resonant to their respective optical transitions, in order to have a well-defined pulse area definition. We discuss local stabilization of the pulse area with or without energy stabilization and as a function of the prolonged distance. Furthermore, we explore some new features for the propagation dynamics in multilevel atomic media which are beyond the traditional framework of McCall and Hahn area theorem and self-induced transparency.
A graph-model is presented to describe multilevel atomic structure. As an example, we take the double L configuration in alkali-metal atoms with hyperfine structure and nuclear spin , as a graph with four vertices. Links are treated as coherence. We introduce the transition matrix which describes the connectivity matrix in static graph-model. In general, the transition matrix describes spatiotemporal behavior of the dynamic graph-model. Furthermore, it describes multiple connections and self-looping of vertices. The atomic excitation is made by short pulses, in order that the hyperfine structure is well resolved. Entropy associated with the proposed dynamic graph-model is used to identify transitions as well as local stabilization in the system without invoking the energy concept of the propagated pulses.
A graph-model is presented to describe multilevel atomic structure. As an example, we take the double Λ configuration in alkali-metal atoms with hyperfine structure and nuclear spin I = 3 / 2 , as a graph with four vertices. Links are treated as coherence. We introduce the transition matrix which describes the connectivity matrix in static graph-model. In general, the transition matrix describes spatiotemporal behavior of the dynamic graph-model. Furthermore, it describes multiple connections and self-looping of vertices. The atomic excitation is made by short pulses, in order that the hyperfine structure is well resolved. Entropy associated with the proposed dynamic graph-model is used to identify transitions as well as local stabilization in the system without invoking the energy concept of the propagated pulses.
In this paper, we establish a graph imaging technique to manifest local stabilization within atomic systems of multiple levels. Specifically, we address the interrelation between local stabilization and image entropy. As an example, we consider the mutual interaction of two pair of pulses propagating in a double-Λ configuration. Thus, we have two different sets of two pulses that share the same shape and phase, initially. The first (second) set belongs to lower (upper) -Λ subsystems, respectively. The configuration of two pair of pulses is considered as a dynamical graph model with four nodes. The dynamic transition matrix describes the connectivity matrix in the static graph model. It is to be emphasized that the graph and its image have the same transition matrix. In particular, the graph model exposes the stabilization in terms of the singular-value decomposition of energies for the transition matrix, that is, irrespectively of the structure of the transition matrix. The image model of the graph displays the details of the matrix structure in terms of row and column probabilities. Therefore, it enables one to study conditional probabilities and mutual information inherent in the network of the graph. Furthermore, the graph imaging provides the main row/column contribution to the transition matrix in terms of image entropy. Our results show that image entropy exposes spatial dependence, which is irrelevant to graph entropy.
We introduce a notion of ``matrix potential'' to nonlinear optical systems. In terms of a matrix potential ggg, we present a gauge-field-theoretic formulation of the Maxwell-Bloch equation that provides a semiclassical description of the propagation of optical pulses through resonant multilevel media. We show that the Bloch part of the equation can be solved identically through ggg and the remaining Maxwell equation becomes a second-order differential equation with a reduced set of variables due to the gauge invariance of the system. Our formulation clarifies the (non-Abelian) symmetry structure of the Maxwell-Bloch equations for various multilevel media in association with symmetric spaces G/HG/HG/H. In particular, we associate the nondegenerate two-level system for self-induced transparency with G/H=SU(2)/U(1)G/H=\mathrm{SU}(2)/\mathrm{U}(1)G/H=SU(2)/U(1) and three-level $\ensuremath{\Lambda}$ or VVV systems with G/H=SU(3)/U(2)G/H=\mathrm{SU}(3)/\mathrm{U}(2)G/H=SU(3)/U(2). We give a detailed analysis for the two-level case in the matrix potential formalism, and address various properties of the system including soliton numbers, effective potential energy, gauge and discrete symmetries, modified pulse area, conserved topological, and nontopological charges. The nontopological charge measures the amount of self-detuning of each pulse. Its conservation law leads to a different type of pulse stability analysis that explains earlier numerical results.