Citation: Khalil, E.M.; Abu-Zinadah, H.; Abd-Rabbou, M.Y. Influence of an External Classical Field on a ♦ Four-Level Atom Inside a Quantized Field. Symmetry 2022, 14, 811. https://doi.org/10.3390/sym14040811 Academic Editors: Vasilis K. Oikonomou, Ignatios Antoniadis and Ioan Ras , a Received: 3 March 2022 Accepted: 11 April 2022 Published: 14 April 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). symmetry S S Article Influence of an External Classical Field on a ♦ Four-Level Atom Inside a Quantized Field Eied Mahmoud Khalil 1 , Hanaa Abu-Zinadah 2 and Mahmoud Youssef Abd-Rabbou 3, * 1 Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia; eiedkhalil@yahoo.com 2 Department of Statistics, College of Science, University of Jeddah, Jeddah 23218, Saudi Arabia; hhabuznadah@uj.edu.sa 3 Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, Egypt * Correspondence: m.elmalky@azhar.edu.eg Abstract: In this paper, we study the effect of detuning parameters and driven external classical field parameters on a quantum system consisting of a ♦-configuration four-level atom inside a quantized cavity field. Under some canonical conditional of dressed states, the exact solution of the Schrödinger equation is obtained. The occupation of atomic levels and statistical population inversion is studied. Our results show that the classical field parameter dissolved the collapse periods and increased the maximum bounds of the upper state, while decreasing the lower bounds of the lower state. The detuning parameters reduce the minimum bounds of atomic levels and their inversion. On the other hand, the linear entropy and l 1 norm of coherence are employed to measure the temporal evolution of the mixedness and coherence. It is found that the driven classical field improves the temporal evolution of the mixedness and lower bounds of coherence. However, the detuning parameters have a destructive effect on the mixedness and lower bounds of coherence. The intensity of the external classical field is regarded as a control parameter with different values of detuning parameters. Keywords: four-level atom; external classical field; mixedness; coherence 1. Introduction The interaction between a quantized field and atomic state is one of the most attractive topics in quantum optics. The Rabi model is the first uncomplicated theoretical model that describes a direct interaction between a single quantized photon and a two-level atom [1]. Another theoretical model that has opened the way for numerous experimental and theoretical quantum studies is called the Jaynes–Cummings model (JCM), which is solved under the rotating wave approximation technique [2]. Overall, the generalization of atom–field interaction has been demonstrated in many theoretical studies, e.g., the interaction between an optical electromagnetic field and N-level atomic stat has been investigated [3,4]. The resonance and non-resonance cases of multi-photon JCM have been discussed [5]. Moreover, high-dimensional atomic states inside an electromagnetic field have been widely studied. For example, the quantum correlation and some statistical characteristics of the three-level atom, four-level atom, and five-level atom have been explored [6–8]. Additionally, the influence of some external effects on atom–field interaction has been proposed, such as Kerr-like medium [9,10], vibrating graphene membrane [11,12], external classical field [13–15], and deformed fields [16,17]. In particular, the interaction between a four-level atom and different types of a cavity mode field has been paid more attention under different configurations [18–22]. The effect of external classical fields in the two-level atom scheme coupled with a quantized electromagnetic field was studied [13]. The squeezing phenomenon and entanglement of JCM in the presence of driven classical field has been discussed [23]. In SU(1,1) Lie algebra, the influence of the off-resonance case Symmetry 2022, 14, 811. https://doi.org/10.3390/sym14040811 https://www.mdpi.com/journal/symmetry