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Study on Graph Entropy Associated with Multilevel Atomic Excitation

Abu Mohamed Alhasan

Published Feb 23, 2021• 0 citations
Publisher

Abstract

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.

DOI
10.9734/bpi/nicst/v8/6541d

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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 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.
Optical dark resonance in systems with arbitrary multilevels and multiphotons is studied theoretically. When the system configuration is treelike (the term of graph theory), all the coherence terms in the system can be categorized into several groups on dark resonance. It is proved for generic system configurations with arbitrary multilevels that the resonance conditions can be obtained successively by setting the determinant of the submatrix of the Hamiltonian with respect to each group to zero. We calculate, as one of the practical examples, all the conditions of dark resonance of potassium atoms with two laser fields and one rf field mixing one excited state and eight ground states in the hyperfine structure. Although the system configuration is very complicated, the condition is easily obtained in an analytic form and verified by the result of a numerical simulation. Furthermore, our theory can reveal the fine structure of dark lines with power broadening in the observed light-induced-fluorescence spectra and numerical calculations. In addition, the dark lines with the complicated splitting pattern observed in experiments can be assigned to the relevant level configuration and transitions by the present theory. \textcopyright{} 1996 The American Physical Society.
The dark-state effect, caused by destructive quantum interference, is an important physical effect in atomic physics and quantum optics. It not only deepens the understanding of light-atom interactions, but also has wide applications in quantum physics and quantum information. Therefore, how to efficiently and conveniently determine the number and form of the dark states in multilevel quantum systems with complex transitions is an important and interesting topic in this field. In this work, we present a general theory for determining the dark states in multilevel quantum systems with any coupling configuration using the arrowhead-matrix method. To confirm the dark states in a multilevel system, we first define the upper- and lower-state subspaces, and then diagonalize the Hamiltonians restricted within the two subspaces to obtain the dressed upper and lower states. By further expressing the transitions between the dressed upper and lower states, we can map the multilevel system to a bipartite-graph network, in which the nodes and links are acted by the dressed states and transitions, respectively. Based on the coupling configurations of the network, we can determine the lower dark states with respect to the upper-state subspace. As examples, we analyze the dark states in three-, four-, and five-level quantum systems, for all possible configurations through the classification of the numbers of upper and lower states. Furthermore, we extend the framework to multilevel quantum systems and discuss the existence of dark states in some typical configurations. We also recover the results of the dark-state polaritons in driven three-level systems with the arrowhead-matrix method. Our theory paves the way for manipulating and utilizing the dark states of multilevel quantum systems in atomic physics and quantum optics.