symmetry SS Article Wigner Function Non-Classicality Induced in a Charge Qubit Interacting with a Dissipative Field Cavity Abdel-Baset A. Mohamed 1,2,* , Eied M. Khalil 3 , Afrah Y. AL-Rezami 1,4 and Hichem Eleuch 5,6,7   Citation: Mohamed, A.-B.A.; Khalil, E.M.; AL-Rezami, A.Y.; Eleuch, H. Wigner Function Non-Classicality Induced in a Charge Qubit Interacting with a Dissipative Field Cavity. Symmetry 2021, 13, 802. https://doi.org/10.3390/sym13050802 Academic Editors: Pedro D. Sacramento, Nikola Paunkovic and Jan Awrejcewicz Received: 20 March 2021 Accepted: 29 April 2021 Published: 4 May 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 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/). 1 Department of Mathematics, College of Science and Humanities in Al-Aflaj, Prince Sattam bin Abdulaziz University, Al-Aflaj 11942, Saudi Arabia; a.mohamed@psau.edu.sa 2 Department of Mathematics, Faculty of Science, Assiut University, Assiut 71515, Egypt 3 Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia; eiedkhalil@tu.edu.sa 4 Department of Statistics and Information, College of Commerce and Economics, Sana’a University, Sana’a 15542, Yemen; a.alrezamee@psau.edu.sa 5 Department of Applied Physics and Astronomy, University of Sharjah, Sharjah 27272, United Arab Emirates; hichem.eleuch@adu.ac.ae 6 Department of Applied Sciences and Mathematics, College of Arts and Sciences, Abu Dhabi University, Abu Dhabi 59911, United Arab Emirates 7 Institute for Quantum Science and Engineering, Texas A&M University, College Station, TX 77843, USA * Correspondence: abdelbastm@aun.edu.eg Abstract: We explore a superconducting charge qubit interacting with a dissipative microwave cavity field. Wigner distribution and its non-classicality are investigated analytically under the effects of the qubit–cavity interaction, the qubit–cavity detuning, and the dissipation. As the microwave cavity field is initially in an even coherent state, we investigate the non-classicality of the Wigner distribu- tions. Partially and maximally frozen entanglement are produced by the qubit–cavity interaction, depending on detuning and cavity dissipation. It is found that the amplitudes and frequency of the Wigner distribution can be controlled by the phase space parameters, the qubit–cavity interaction and the detuning, as well as by the dissipation. The cavity dissipation reduces the non-classicality; this process can be accelerated by the detuning. Keywords: charge–qubit system; quasi-probability wigner function; entanglement 1. Introduction The decoherence and dissipation issues taint the dynamics of every quantum system. These effects reduce, distort or destroy the quantum phenomena, [1,2] such as: quan- tum coherence, squeezing, and quantum correlation. Moreover, decoherence is the most significant characteristic of an open quantum system. This quantum effect destroys the nonclassical correlations. Decoherence affects the entangled states and transforms them to mixed states [3]. Decoherence usually occurs as the system’s constituents interact with the environment [4,5]. The effects of decoherence and dissipations on the dynamical features have been investigated in various quantum systems [6–9]. The interaction between the quantum systems and the environment usually leads to a decoherence/dissipation process, which reduces the quantum phenomena [10]. The decoherence and the dissipation effects might lead to spontaneous symmetry breaking or phase transition phenomena [11–15], which may occur in several dissipative quantum systems [16–19]. These effects erase the quantum information resources. In general, the decoherence and the dissipation effects can be investigated by various types of master equation [20–24] which can be employed to analyze the quantum dynamics of the systems. To characterize quantum states and present valuable quantum information about the system states, quasi-probability distributions were introduced [25]. Wigner distribu- Symmetry 2021, 13, 802. https://doi.org/10.3390/sym13050802 https://www.mdpi.com/journal/symmetry