{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T17:02:45Z","timestamp":1761930165747,"version":"build-2065373602"},"reference-count":67,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,2,22]],"date-time":"2021-02-22T00:00:00Z","timestamp":1613952000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We consider the interaction of a qubit system with a two-mode field in the presence of multi-photon transition and phase damping effect. We use the master equation to obtain the density operator when the qubit is initially prepared in its excited state and the field is in a finite-dimensional pair coherent state. The properties of the considered system, such as the population inversion, amount of the mixedness, parameter estimation, and squeezing, are explored for one- and two-photon transitions. The effects of photon addition to the field and phase damping on the evaluation of these quantumness measures are also investigated.<\/jats:p>","DOI":"10.3390\/e23020252","type":"journal-article","created":{"date-parts":[[2021,2,22]],"date-time":"2021-02-22T14:23:27Z","timestamp":1614003807000},"page":"252","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Dephasing Process of a Single Atom Interacting with a Two-Mode Field"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0189-845X","authenticated-orcid":false,"given":"Eied M.","family":"Khalil","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kamal","family":"Berrada","sequence":"additional","affiliation":[{"name":"Physics Department, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sayed","family":"Abdel-Khalek","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Beida","family":"Alsubei","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4596-137X","authenticated-orcid":false,"given":"Hichem","family":"Eleuch","sequence":"additional","affiliation":[{"name":"Department of Applied Physics and Astronomy, University of Sharjah, Sharjah27272, United Arab Emirates"},{"name":"Department of Applied Sciences and Mathematics, College of Arts and Sciences, Abu Dhabi University, Abu Dhabi 59911, United Arab Emirates"},{"name":"Institute for Quantum Science and Engineering, Texas A&amp;M University, College Station, Texas, TX 77843, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,2,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1109\/PROC.1963.1664","article-title":"Comparison of quantum and semiclassical radiation theories with application to the beam maser","volume":"51","author":"Jaynes","year":"1963","journal-title":"Proc. IEEE"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1103\/PhysRevLett.58.353","article-title":"Observation of quantum collapse and revival in a one-atom maser","volume":"57","author":"Rempe","year":"1987","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"047905","DOI":"10.1103\/PhysRevLett.90.047905","article-title":"Entanglement and entropy engineering of atomic two-qubit states","volume":"90","author":"Clark","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"043812","DOI":"10.1103\/PhysRevA.62.043812","article-title":"Nonlinear dynamical evolution of the driven two-photon Jaynes-Cummings model","volume":"62","author":"Joshi","year":"2000","journal-title":"Phys. Rev. A"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1323","DOI":"10.1103\/PhysRevLett.44.1323","article-title":"Periodic Spontaneous Collapse and Revival in a Simple Quantum Model","volume":"44","author":"Eberly","year":"1980","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1103\/PhysRevLett.58.203","article-title":"Nonclassical radiation of a single stored ion","volume":"58","author":"Diedrich","year":"1987","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"966","DOI":"10.1103\/PhysRevLett.50.966","article-title":"Chaos in the semiclassical N-atom Jaynes-Cummings model: Failure of the rotating-wave approximation","volume":"50","author":"Milonni","year":"1983","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3175","DOI":"10.1103\/PhysRevA.37.3175","article-title":"Strong squeezing in the Jaynes-Cummings model","volume":"37","author":"Kuklinski","year":"1988","journal-title":"Phys. Rev. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"934","DOI":"10.1103\/PhysRevLett.63.934","article-title":"Harmonic oscillator driven by a quantum current","volume":"63","author":"Slosser","year":"1990","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","unstructured":"Nielsen, M.A., and Chuang, I.L. (2000). Quantum Computation and Quantum Information, Cambridge University Press."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"042315","DOI":"10.1103\/PhysRevA.64.042315","article-title":"Universal state inversion and concurrence in arbitrary dimensions","volume":"64","author":"Rungta","year":"2001","journal-title":"Phys. Rev. A"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Benenti, G., Casati, G., and Strini, G. (2004). Principle of Quantum Computation and Information, World Scientific.","DOI":"10.1142\/9789812794796"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"935","DOI":"10.1016\/0378-4371(90)90300-H","article-title":"Collective spontaneous radiation of two atoms in the finite-Q cavity","volume":"167","author":"Bashkirov","year":"1990","journal-title":"Physica A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1016\/0030-4018(89)90367-2","article-title":"Simultaneous jumps in two two-level atoms","volume":"70","author":"Kim","year":"1989","journal-title":"Opt. Commun."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3059","DOI":"10.1103\/PhysRevA.57.3059","article-title":"Cavity-induced two-photon absorption in unidentical atoms","volume":"57","author":"Kim","year":"1998","journal-title":"Phys. Rev. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"040101","DOI":"10.1103\/PhysRevA.65.040101","article-title":"Entanglement induced by a single-mode heat environment","volume":"65","author":"Kim","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2854","DOI":"10.1016\/j.optcom.2008.01.051","article-title":"Single-atom entropy squeezing for two two-level atoms interacting with a single-mode radiation field","volume":"281","author":"Wahiddin","year":"2008","journal-title":"Opt. Commun."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2247","DOI":"10.1007\/s10773-006-9191-3","article-title":"Entangling two-atom through cooperative interaction under stimulated emission","volume":"45","author":"Guo","year":"2006","journal-title":"Int. J. Theor. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3101","DOI":"10.1007\/s10773-008-9743-9","article-title":"Stationary state entanglement of two atoms inside an optical cavity under noise","volume":"47","author":"Ozel","year":"2008","journal-title":"Int. J. Theor. Phys."},{"key":"ref_20","unstructured":"Von Neumann, J. (1932). Mathematische Grundlagen der Quantenmechanik, Springer."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1941007","DOI":"10.1142\/S0219749919410077","article-title":"Entropy fluctuations to quantify mixedness","volume":"17","year":"2019","journal-title":"Int. J. Quantum Inf."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"5132","DOI":"10.1103\/PhysRevA.41.5132","article-title":"Wave-packet evolution in the damped oscillator","volume":"41","author":"Phoenix","year":"1990","journal-title":"Phys. Rev. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1016\/0003-4916(88)90006-1","article-title":"Fluctuations and entropy in models of quantum optical resonance","volume":"186","author":"Phoenix","year":"1988","journal-title":"Ann. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.pquantelec.2017.07.002","article-title":"Journeys from quantum optics to quantum technology","volume":"54","author":"Barnett","year":"2017","journal-title":"Progr. Quant. Elec."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Pathak, A. (2013). Elements of Quantum Computation and Quantum Communication, CRC Press.","DOI":"10.1201\/b15007"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1103\/PhysRevA.50.67","article-title":"Squeezed atomic states and projection noise in spectroscopy","volume":"50","author":"Wineland","year":"1994","journal-title":"Phys. Rev. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"5138","DOI":"10.1103\/PhysRevA.47.5138","article-title":"Squeezed spin states","volume":"47","author":"Kitagawa","year":"1993","journal-title":"Phys. Rev. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1103\/PhysRevLett.47.709","article-title":"Reduced quantum fluctuations in resonance fluorescence","volume":"47","author":"Walls","year":"1981","journal-title":"Phys. Rev. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"3487","DOI":"10.1103\/PhysRevLett.80.3487","article-title":"Quantum noise of an atomic spin polarization measurement","volume":"80","author":"Sorensen","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"4750","DOI":"10.1103\/PhysRevB.32.4750","article-title":"Reduced quantum fluctuations in the Josephson junction","volume":"32","author":"Wodkiewicz","year":"1985","journal-title":"Phys. Rev. B"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"8121","DOI":"10.1103\/PhysRevA.45.8121","article-title":"Atomic-dipole squeezing and emission spectra of the nondegenerate two-photon Jaynes-Cummings model","volume":"45","author":"Ashraf","year":"1992","journal-title":"Phys. Rev. A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"4782","DOI":"10.1103\/PhysRevLett.79.4782","article-title":"Spin squeezing in an ensemble of atoms illuminated with squeezed light","volume":"79","author":"Kuzmicah","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3869","DOI":"10.1103\/PhysRevLett.79.3869","article-title":"Quantum-controlled few-photon state generated by squeezed atoms","volume":"79","author":"Sairo","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"724","DOI":"10.1038\/s41566-018-0301-6","article-title":"Advances in photonic quantum sensing","volume":"12","author":"Pirandola","year":"2018","journal-title":"Nat. Photon."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3865","DOI":"10.1103\/PhysRevLett.79.3865","article-title":"Improvement of frequency standards with quantum entanglement","volume":"79","author":"Huelga","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1330","DOI":"10.1126\/science.1104149","article-title":"Quantum-enhanced measurements: Beating the standard quantum limit","volume":"306","author":"Giovannetti","year":"2004","journal-title":"Science"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"010401","DOI":"10.1103\/PhysRevLett.96.010401","article-title":"Quantum metrology","volume":"96","author":"Giovannetti","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1038\/nphoton.2011.35","article-title":"Advances in quantum metrology","volume":"5","author":"Giovannetti","year":"2011","journal-title":"Nat. Photon."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1055","DOI":"10.1007\/s11082-013-9822-x","article-title":"Quantum Fisher information flow and entanglement in pair coherent states","volume":"46","year":"2014","journal-title":"Opt. Quantum Eletronics"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"233601","DOI":"10.1103\/PhysRevLett.109.233601","article-title":"Quantum metrology in non-Markovian environments","volume":"109","author":"Chin","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1063","DOI":"10.1038\/ncomms2067","article-title":"The elusive Heisenberg limit in quantum-enhanced metrology","volume":"3","year":"2012","journal-title":"Nat. Commun."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"120401","DOI":"10.1103\/PhysRevLett.111.120401","article-title":"Noisy metrology beyond the standard quantum limit","volume":"111","author":"Chaves","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"035806","DOI":"10.1103\/PhysRevA.88.035806","article-title":"Non-Markovian effect on the precision of parameter estimation","volume":"88","author":"Berrada","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"7282","DOI":"10.1038\/ncomms8282","article-title":"Robust quantum metrological schemes based on protection of quantum Fisher information","volume":"6","author":"Lu","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"052105","DOI":"10.1103\/PhysRevA.91.052105","article-title":"Classical-driving-enhanced parameter-estimation precision of a non-Markovian dissipative two-state system","volume":"91","author":"Li","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1364\/JOSAB.32.000571","article-title":"Protecting the precision of estimation in a photonic crystal","volume":"32","author":"Berrada","year":"2015","journal-title":"Opt. Soc. Am. B"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"012117","DOI":"10.1103\/PhysRevA.96.012117","article-title":"Quantum parameter estimation with optimal control","volume":"96","author":"Liu","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Breuer, H.-P., and Petruccione, F. (2002). The Theory of Open Quantum Systems, Oxford University Press.","DOI":"10.1007\/3-540-44874-8_4"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"085503","DOI":"10.1088\/0953-4075\/43\/8\/085503","article-title":"Controllable atom\u2013photon entanglement near a 3D anisotropic photonic band edge","volume":"43","author":"Entezar","year":"2010","journal-title":"Phys. B Atom. Mol. Opt. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"085501","DOI":"10.1088\/0953-4075\/44\/8\/085501","article-title":"The effect of a coupling field on the entanglement dynamics of a three-level atom","volume":"44","author":"Mortezapour","year":"2011","journal-title":"Phys. B Atom. Mol. Opt. Phys."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"060310","DOI":"10.1088\/1674-1056\/22\/6\/060310","article-title":"Phase-controlled atom\u2014photon entanglement in a three-level \u039b\u2014type closed-loop atomic system","volume":"22","author":"Mortezapour","year":"2013","journal-title":"Chin. Phys. B"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Abazari, M., Mortezapour, A., Mahmoudi, M., and Sahrai, M. (2011). Phase-controlled atom-photon entanglement in a three-level V-type atomic system via spontaneously generated coherence. Entropy, 13.","DOI":"10.3390\/e13091541"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"813","DOI":"10.2478\/s11534-014-0520-5","article-title":"The influence of spontaneously generated coherence on atom-photon entanglement in a \u039b-type system with an incoherent pump","volume":"12","author":"Yang","year":"2014","journal-title":"Open Phys."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"168073","DOI":"10.1016\/j.aop.2020.168073","article-title":"Quantumness and memory of one qubit in a dissipative cavity under classical control","volume":"441","author":"Gholipour","year":"2020","journal-title":"Ann. Phys."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"050504","DOI":"10.1103\/PhysRevLett.98.050504","article-title":"Remote preparation of an atomic quantum memory","volume":"98","author":"Rosenfeld","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"5932","DOI":"10.1103\/PhysRevLett.81.5932","article-title":"Quantum repeaters: The role of imperfect local operations in quantum communication","volume":"81","author":"Briegel","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1098","DOI":"10.1038\/nature07241","article-title":"Experimental demonstration of a BDCZ quantum repeater node","volume":"454","author":"Yuan","year":"2008","journal-title":"Nature"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"014032","DOI":"10.1088\/0031-8949\/2009\/T135\/014032","article-title":"Entanglement sudden death and sudden birth in two uncoupled spins","volume":"135","author":"Ferraro","year":"2009","journal-title":"Phys. Scr. T."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"045323","DOI":"10.1103\/PhysRevB.70.045323","article-title":"Non-Markovian dynamics in a spin star system: Exact solution and approximation techniques","volume":"70","author":"Breuer","year":"2004","journal-title":"Phys. Rev. B"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"042312","DOI":"10.1103\/PhysRevA.69.042312","article-title":"Mediated entanglement and correlations in a star network of interacting spins","volume":"69","author":"Hutton","year":"2003","journal-title":"Phys. Rev. A"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"035302","DOI":"10.1088\/1751-8113\/44\/3\/035302","article-title":"Variance squeezing and entanglement of the XX central spin model","volume":"44","year":"2011","journal-title":"Phys. A Math. Theor."},{"key":"ref_62","unstructured":"Biedenharn, L., and van Dam, H. (1965). Quantum Theory of Angular Momentum, Academic Press."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.optcom.2005.09.080","article-title":"Generation and some non-classical properties of a finite dimensional pair coherent state","volume":"260","author":"Obada","year":"2006","journal-title":"Opt. Commun."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"064103","DOI":"10.1103\/PhysRevA.80.064103","article-title":"Quantum master equation for dephasing of a two-level system with an initial correlation","volume":"80","author":"Ban","year":"2009","journal-title":"Phys. Rev. A"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"1103","DOI":"10.1103\/PhysRevLett.60.1103","article-title":"Generalized entropic uncertainty relations","volume":"60","author":"Maassen","year":"1988","journal-title":"Phys. Rev. Lett."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"1043","DOI":"10.1080\/09500340008233404","article-title":"Entropy squeezing for a two-level atom","volume":"47","author":"Fang","year":"2000","journal-title":"J. Mod. Optics"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1016\/j.chaos.2006.06.067","article-title":"Entropy squeezing of time dependent single-mode Jaynes\u2013Cummings model in presence of non-linear effect","volume":"36","author":"Abdalla","year":"2008","journal-title":"Chaos Solitons Fractals"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/2\/252\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:28:12Z","timestamp":1760160492000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/2\/252"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,22]]},"references-count":67,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["e23020252"],"URL":"https:\/\/doi.org\/10.3390\/e23020252","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2021,2,22]]}}}