a Department of Mathematics, Kuztown University of Pennsylvania,15200 Kutztown Road, Kuztown, PA-... more a Department of Mathematics, Kuztown University of Pennsylvania,15200 Kutztown Road, Kuztown, PA-19530, USA b Department of Mathematics, Faculty of Science for Girls, King Abdulaziz University, Jeddah, Saudi Arabia c Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah-21589, Saudi Arabia d Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt e School of Electronics and Information Engineering, Wuhan Donghu University, Wuhan-430212, P. R. China f School of Physics and Technology, Wuhan University, Wuhan-430072, P.R. China g Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA
In this paper we propose a novel computational algorithm for solving ordinary differential equati... more In this paper we propose a novel computational algorithm for solving ordinary differential equations with non-constants coefficients by using the modified version of Laplace and Sumudu transforms which is called Elzaki transform. Elzaki transform can be easily applied to the initial value problems with less computational work. The several illustrative examples can not solve by Sumudu transform, this means that Elzaki transform is a powerful tool for solving some ordinary differential equations with variable coefficients.
In this paper, we apply modified version of double Sumudu transform which is called double Elzaki... more In this paper, we apply modified version of double Sumudu transform which is called double Elzaki transform to solve the general linear telegraph equation. The applicability of this new transform is demonstrated using some functions, which arise in the solution of general linear telegraph equation.
We study the interaction of a three two-level atoms (3-2LA) with a one-mode op?tical coherent fie... more We study the interaction of a three two-level atoms (3-2LA) with a one-mode op?tical coherent field in coherent state in the presence of non-linear Kerr medium. The three atoms are initially prepared in upper and entangled states while the field mode is in a coherent state. The constants of motion, 3-2LA and field density matrix are obtained. The analytic results are employed to perform some investigations of the temporal evolution of the von Neumann entropy as measure of the degree of entanglement between the 3-2LA and optical coherent field. The effect of the detuning and the initial atomic states on the evolution of geometric phase and en?tanglement is analyzed. Also, we demonstrate the link between the geometric phase and non-classical properties during the evolution time. Additionally the effect of detuning and initial conditions on the Mandel parameter is studied. The obtained results are emphasize the impact of the detuning and the initial atomic states of the feature of the ...
The effect of the field-field interaction on a cavity containing two qubit (TQ) interacting with ... more The effect of the field-field interaction on a cavity containing two qubit (TQ) interacting with a two mode of electromagnetic field as parametric amplifier type is investigated. After performing an appropriate transformation, the constants of motion are calculated. Using the Schrödinger differential equation a system of differential equations was obtained, and the general solution was obtained in the case of exact resonance. Some statistical quantities were calculated and discussed in detail to describe the features of this system. The collapses and revivals phenomena have been discussed in details. The Shannon information entropy has been applied for measuring the degree of entanglement (DE) between the qubits and the electromagnetic field. The normal squeezing for some values of the parameter of the field-field interaction is studied. The results showed that the collapses disappeared after the field-field terms were added and the maximum values of normal squeezing decrease when increasing of the field-field interaction parameter. While the revivals and amplitudes of the oscillations increase when the parameter of the field-field interaction increases. Degree of entanglement is partially more entangled with increasing of the field-field interaction parameter. The relationship between revivals, collapses and the degree of entanglement (Shannon information entropy) was monitored and discussed in the presence and absence of the field-field interaction.
In this work, we introduce a model of a two-atom interacting with a multi-level atom governed by ... more In this work, we introduce a model of a two-atom interacting with a multi-level atom governed by su(2) Lie algebra in the presence of external classical field. The influence of the classical field on the system is discussed in detail for certain values for the classical terms. The atomic density matrix of the proposed system is obtained. The dynamical behavior of the atomic Fisher information as an indicator of the nonlocal correlation between a two-atom and su(2) field is discussed. Moreover, we examine the effect of classical field on the evolution of entropy squeezing and the geometric phase induced between the initial and final state of the proposed system. The results outlined some important phenomena as sudden death and sudden birth of entanglement in presence of the classical terms is observed through the dynamics of atomic Fisher information in the presence of classical field for the large number of levels.
We present a detail study of the evolution of nonlocal correlations of an interacting quantum sys... more We present a detail study of the evolution of nonlocal correlations of an interacting quantum system comprising a three-level atom and a field mode initially prepared in a squeezed vacuum state with added photons. We compare the dynamical behavior of the quantum phase and entanglement by varying the number of photons added to the squeezed vacuum state. Furthermore, we examine the influence of the added-photon number and the squeeze parameter on the dynamical behavior of entanglement, quantum phase, and nonclassical properties of the field. Moreover, we explore the link between the quantum phase and the nonlocal correlation. Finally, we introduce an effective method to generate and maintain a high level of entanglement for this quantum system based on precise parameter ranges.
We consider a system consisting of two two-level atoms interacting with a radiation field, includ... more We consider a system consisting of two two-level atoms interacting with a radiation field, including Stark shift terms, and investigate the effect of Stark shift terms on the interaction between the radiation field and the two atoms. Within the framework of the Heisenberg picture, we obtain the general solution to the operator equations of motion. In addition, we derive the general solution obtained by solving the system of differential equations. Some statistical aspects such as atomic inversion and linear entropy are discussed in detail. We study the effect of the time-dependent function on the population inversion and linear entropy. Finally, we examine the linear entropy, concurrence, and quantum and classical correlations for different values of the detuning parameter.
Abstract In this paper, we study the interaction between a two-level atom and a three types of no... more Abstract In this paper, we study the interaction between a two-level atom and a three types of nonlinear interaction of two modes, two as a degenerate two photon and the third of the converter form. The solution to the Schrodinger equation is obtained exactly under a certain specific condition. By using this solution we discuss numerically the atomic inversion, the degree of entanglement through the atomic Wehrl entropy, the geometric phase entropy for chosen values of the detuning and coupling parameters. Statistical properties of the investigated system is discussed by the evolution of the Mandel parameter.
In this paper, we develop the model of the four-level double Raman pairs by exploiting the requir... more In this paper, we develop the model of the four-level double Raman pairs by exploiting the required optimal conditions for this system that are feasible with real experimental realization. We investigate qualitatively the entanglement, statistical properties, and geometric phase for the pair of Stokes and anti-Stokes photons in the presence of the time-dependent coupling effect. We show that these quantifiers are very sensitive to the change of the Rabi frequency and time, exhibiting substantial phenomena that are depending on the kind of coupling between the atom and photons. Finally, we explore the relationship between the quantum quantifiers in terms of the physical parameters with and without timedependent coupling effect.
In this work, we present a reliable combination of homotopy perturbation method and Elzaki transf... more In this work, we present a reliable combination of homotopy perturbation method and Elzaki transform to investigate some nonlinear partial differential equations. The nonlinear terms can be handled by the use of homotopy perturbation method. The proposed homotopy perturbation method is applied to the reformulated first and second order initial value problem which leads the solution in terms of transformed variables, and the series solution is obtained by making use of the inverse transformation. The results show the efficiency of this method.
In this paper, we apply modified version of double Sumudu transform which is called double Elzaki... more In this paper, we apply modified version of double Sumudu transform which is called double Elzaki transform to solve the general linear telegraph equation. The applicability of this new transform is demonstrated using some functions, which arise in the solution of general linear telegraph equation.
The aim of this study is to solve some linear and nonlinear partial differential equations using ... more The aim of this study is to solve some linear and nonlinear partial differential equations using the new integral transform "Elzaki transform" and projected differential transform method. The nonlinear terms can be handled by using of projected differential transform method; this method is more efficient and easy to handle such partial differential equations in comparison to other methods. The results show the efficiency and validation of this method.
In this work modified of Sumudu transform [10,11,12] which is called Elzaki transform method (new... more In this work modified of Sumudu transform [10,11,12] which is called Elzaki transform method (new integral transform) is considered to solve general linear telegraph equation, this method is a powerful tool for solving differential equations and integral equations [1, 2, 3, 4, 5]. Using modified of Sumudu transform or Elzaki transform, it is possible to find the exact solution of telegraph equation. This method is more efficient and easier to handle as compare to the Sumudu transform method and variational iteration method. To illustrate the ability of the method some examples are provided.
We investigate some existence and stability results for the Darboux problem of partial fractional... more We investigate some existence and stability results for the Darboux problem of partial fractional random differential equations in Banach spaces. Our existence results are based upon some fixed point theorems.
The polarization dependence of the leptonic -pairs production due to the photon interaction with ... more The polarization dependence of the leptonic -pairs production due to the photon interaction with the coulomb field of light nuclei, have been studied in details. The circular polarization of the photon and the longitudinal polarization of leptons have been taken into account obtaining general formulas for the energy distribution cross-section of the pair-production process, containing terms proportional to the polarization correlation between photon and particles , and between the particles of the pair . Obtaining also general formulas for the relative probability of the pair production in two cases of spin correlation . Showing that the study of the photon and particles polarization increases the values of cross-sections by very large amounts and consequently increases the probability of occurring the pair-production process.
a Department of Mathematics, Kuztown University of Pennsylvania,15200 Kutztown Road, Kuztown, PA-... more a Department of Mathematics, Kuztown University of Pennsylvania,15200 Kutztown Road, Kuztown, PA-19530, USA b Department of Mathematics, Faculty of Science for Girls, King Abdulaziz University, Jeddah, Saudi Arabia c Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah-21589, Saudi Arabia d Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt e School of Electronics and Information Engineering, Wuhan Donghu University, Wuhan-430212, P. R. China f School of Physics and Technology, Wuhan University, Wuhan-430072, P.R. China g Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA
In this paper we propose a novel computational algorithm for solving ordinary differential equati... more In this paper we propose a novel computational algorithm for solving ordinary differential equations with non-constants coefficients by using the modified version of Laplace and Sumudu transforms which is called Elzaki transform. Elzaki transform can be easily applied to the initial value problems with less computational work. The several illustrative examples can not solve by Sumudu transform, this means that Elzaki transform is a powerful tool for solving some ordinary differential equations with variable coefficients.
In this paper, we apply modified version of double Sumudu transform which is called double Elzaki... more In this paper, we apply modified version of double Sumudu transform which is called double Elzaki transform to solve the general linear telegraph equation. The applicability of this new transform is demonstrated using some functions, which arise in the solution of general linear telegraph equation.
We study the interaction of a three two-level atoms (3-2LA) with a one-mode op?tical coherent fie... more We study the interaction of a three two-level atoms (3-2LA) with a one-mode op?tical coherent field in coherent state in the presence of non-linear Kerr medium. The three atoms are initially prepared in upper and entangled states while the field mode is in a coherent state. The constants of motion, 3-2LA and field density matrix are obtained. The analytic results are employed to perform some investigations of the temporal evolution of the von Neumann entropy as measure of the degree of entanglement between the 3-2LA and optical coherent field. The effect of the detuning and the initial atomic states on the evolution of geometric phase and en?tanglement is analyzed. Also, we demonstrate the link between the geometric phase and non-classical properties during the evolution time. Additionally the effect of detuning and initial conditions on the Mandel parameter is studied. The obtained results are emphasize the impact of the detuning and the initial atomic states of the feature of the ...
The effect of the field-field interaction on a cavity containing two qubit (TQ) interacting with ... more The effect of the field-field interaction on a cavity containing two qubit (TQ) interacting with a two mode of electromagnetic field as parametric amplifier type is investigated. After performing an appropriate transformation, the constants of motion are calculated. Using the Schrödinger differential equation a system of differential equations was obtained, and the general solution was obtained in the case of exact resonance. Some statistical quantities were calculated and discussed in detail to describe the features of this system. The collapses and revivals phenomena have been discussed in details. The Shannon information entropy has been applied for measuring the degree of entanglement (DE) between the qubits and the electromagnetic field. The normal squeezing for some values of the parameter of the field-field interaction is studied. The results showed that the collapses disappeared after the field-field terms were added and the maximum values of normal squeezing decrease when increasing of the field-field interaction parameter. While the revivals and amplitudes of the oscillations increase when the parameter of the field-field interaction increases. Degree of entanglement is partially more entangled with increasing of the field-field interaction parameter. The relationship between revivals, collapses and the degree of entanglement (Shannon information entropy) was monitored and discussed in the presence and absence of the field-field interaction.
In this work, we introduce a model of a two-atom interacting with a multi-level atom governed by ... more In this work, we introduce a model of a two-atom interacting with a multi-level atom governed by su(2) Lie algebra in the presence of external classical field. The influence of the classical field on the system is discussed in detail for certain values for the classical terms. The atomic density matrix of the proposed system is obtained. The dynamical behavior of the atomic Fisher information as an indicator of the nonlocal correlation between a two-atom and su(2) field is discussed. Moreover, we examine the effect of classical field on the evolution of entropy squeezing and the geometric phase induced between the initial and final state of the proposed system. The results outlined some important phenomena as sudden death and sudden birth of entanglement in presence of the classical terms is observed through the dynamics of atomic Fisher information in the presence of classical field for the large number of levels.
We present a detail study of the evolution of nonlocal correlations of an interacting quantum sys... more We present a detail study of the evolution of nonlocal correlations of an interacting quantum system comprising a three-level atom and a field mode initially prepared in a squeezed vacuum state with added photons. We compare the dynamical behavior of the quantum phase and entanglement by varying the number of photons added to the squeezed vacuum state. Furthermore, we examine the influence of the added-photon number and the squeeze parameter on the dynamical behavior of entanglement, quantum phase, and nonclassical properties of the field. Moreover, we explore the link between the quantum phase and the nonlocal correlation. Finally, we introduce an effective method to generate and maintain a high level of entanglement for this quantum system based on precise parameter ranges.
We consider a system consisting of two two-level atoms interacting with a radiation field, includ... more We consider a system consisting of two two-level atoms interacting with a radiation field, including Stark shift terms, and investigate the effect of Stark shift terms on the interaction between the radiation field and the two atoms. Within the framework of the Heisenberg picture, we obtain the general solution to the operator equations of motion. In addition, we derive the general solution obtained by solving the system of differential equations. Some statistical aspects such as atomic inversion and linear entropy are discussed in detail. We study the effect of the time-dependent function on the population inversion and linear entropy. Finally, we examine the linear entropy, concurrence, and quantum and classical correlations for different values of the detuning parameter.
Abstract In this paper, we study the interaction between a two-level atom and a three types of no... more Abstract In this paper, we study the interaction between a two-level atom and a three types of nonlinear interaction of two modes, two as a degenerate two photon and the third of the converter form. The solution to the Schrodinger equation is obtained exactly under a certain specific condition. By using this solution we discuss numerically the atomic inversion, the degree of entanglement through the atomic Wehrl entropy, the geometric phase entropy for chosen values of the detuning and coupling parameters. Statistical properties of the investigated system is discussed by the evolution of the Mandel parameter.
In this paper, we develop the model of the four-level double Raman pairs by exploiting the requir... more In this paper, we develop the model of the four-level double Raman pairs by exploiting the required optimal conditions for this system that are feasible with real experimental realization. We investigate qualitatively the entanglement, statistical properties, and geometric phase for the pair of Stokes and anti-Stokes photons in the presence of the time-dependent coupling effect. We show that these quantifiers are very sensitive to the change of the Rabi frequency and time, exhibiting substantial phenomena that are depending on the kind of coupling between the atom and photons. Finally, we explore the relationship between the quantum quantifiers in terms of the physical parameters with and without timedependent coupling effect.
In this work, we present a reliable combination of homotopy perturbation method and Elzaki transf... more In this work, we present a reliable combination of homotopy perturbation method and Elzaki transform to investigate some nonlinear partial differential equations. The nonlinear terms can be handled by the use of homotopy perturbation method. The proposed homotopy perturbation method is applied to the reformulated first and second order initial value problem which leads the solution in terms of transformed variables, and the series solution is obtained by making use of the inverse transformation. The results show the efficiency of this method.
In this paper, we apply modified version of double Sumudu transform which is called double Elzaki... more In this paper, we apply modified version of double Sumudu transform which is called double Elzaki transform to solve the general linear telegraph equation. The applicability of this new transform is demonstrated using some functions, which arise in the solution of general linear telegraph equation.
The aim of this study is to solve some linear and nonlinear partial differential equations using ... more The aim of this study is to solve some linear and nonlinear partial differential equations using the new integral transform "Elzaki transform" and projected differential transform method. The nonlinear terms can be handled by using of projected differential transform method; this method is more efficient and easy to handle such partial differential equations in comparison to other methods. The results show the efficiency and validation of this method.
In this work modified of Sumudu transform [10,11,12] which is called Elzaki transform method (new... more In this work modified of Sumudu transform [10,11,12] which is called Elzaki transform method (new integral transform) is considered to solve general linear telegraph equation, this method is a powerful tool for solving differential equations and integral equations [1, 2, 3, 4, 5]. Using modified of Sumudu transform or Elzaki transform, it is possible to find the exact solution of telegraph equation. This method is more efficient and easier to handle as compare to the Sumudu transform method and variational iteration method. To illustrate the ability of the method some examples are provided.
We investigate some existence and stability results for the Darboux problem of partial fractional... more We investigate some existence and stability results for the Darboux problem of partial fractional random differential equations in Banach spaces. Our existence results are based upon some fixed point theorems.
The polarization dependence of the leptonic -pairs production due to the photon interaction with ... more The polarization dependence of the leptonic -pairs production due to the photon interaction with the coulomb field of light nuclei, have been studied in details. The circular polarization of the photon and the longitudinal polarization of leptons have been taken into account obtaining general formulas for the energy distribution cross-section of the pair-production process, containing terms proportional to the polarization correlation between photon and particles , and between the particles of the pair . Obtaining also general formulas for the relative probability of the pair production in two cases of spin correlation . Showing that the study of the photon and particles polarization increases the values of cross-sections by very large amounts and consequently increases the probability of occurring the pair-production process.
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