# Entropy Associated with Information Storage and Its Retrieval

## Abstract

**:**

## 1. Introduction

## 2. The Theoretical Description

**Figure 1.**Excitation of the sodium ${D}_{1}$ line. The dashed arrow shows the excitation with one beam, which is tuned to the fine structure transition. $\text{\Delta}{\omega}_{0}$ is the detuning. The solid arrows show the double-Λ configuration excitation with ${\text{\Omega}}_{p}$ and ${\text{\Omega}}_{c}$ fields. ${\text{\Delta}}_{p}$ and ${\text{\Delta}}_{c}$ are the detuning of the fields with ${\text{\Omega}}_{p}$ and ${\text{\Omega}}_{c}$ from the transitions $|1\rangle \to |3\rangle $ and $|2\rangle \to |3\rangle $, respectively. $\text{\Delta}{\omega}_{2,1}$ and $\text{\Delta}{\omega}_{4,3}$ are lower and upper hyperfine splitting, respectively.

## 3. Spectral Dependence of Entropies Associated with Light Scattering for Single Color Excitation of ${\mathbf{D}}_{\mathbf{1}}$ Line

**Figure 2.**Spectral dependence of the probability cross-sections for the Rayleigh and π scattered light at different atom-field couplings: weak, moderate and strong.

**Figure 3.**The dependence of relative entropy on the atom-field coupling v for: the scattered π polarized light ${S}_{\pi}(\Omega ;v)$, Rayleigh scattering ${S}_{Ry}(\Omega ;v)$ and Shannon entropy, based on populations of the hyperfine levels ${S}_{n}(\Omega ;v)$.

## 4. Time-Dependent Entropy Associated with Light Storage and Light Restoring

**Figure 4.**Space dependence of the relative Shannon entropy associated with ${V}_{s}$ and ${V}_{r}$ subsystems. The inset shows the space dependence of relative Shannon entropy for the period $T={T}_{s}\cup {T}_{r}$ .

**Figure 5.**The space dependence of the relative entropies ${H}_{p}$ and ${H}_{c}$. The probability is defined as the ratio of energy of the fields in the transitions $|1\rangle \leftrightarrow |3\rangle $ and $|2\rangle \leftrightarrow |3\rangle $ to the corresponding total field energy in each transition, respectively.

**Figure 6.**The space dependence of the cumulative sum ${S}_{e}\left(z\right)$ of the relative Shannon entropy ${H}_{e}$ associated with the probability distribution ${P}_{e}\left(z\right)$, where $e=\{p,r,{c}_{1},{c}_{2}\}$ stand for the p-field, restoring field, the first coupling field and the second coupling field, respectively.

## 5. Conclusions

## Conflicts of Interest

## Appendix: The Time Evolution of the Density Matrix Equations

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Alhasan, A.M.
Entropy Associated with Information Storage and Its Retrieval. *Entropy* **2015**, *17*, 5920-5937.
https://doi.org/10.3390/e17085920

**AMA Style**

Alhasan AM.
Entropy Associated with Information Storage and Its Retrieval. *Entropy*. 2015; 17(8):5920-5937.
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**Chicago/Turabian Style**

Alhasan, Abu Mohamed.
2015. "Entropy Associated with Information Storage and Its Retrieval" *Entropy* 17, no. 8: 5920-5937.
https://doi.org/10.3390/e17085920