## Some of A-S F Obada's publications

- R K Bullough, A-S F Obada, B V Thompson and F Hynne, Microscopic Theory of the Einstein optical scattering equations,
*Chemical Physical Letters***2**(5) (1968), 293-296.We have obtained the phenomenological scattering formulae of Einstein by microscopic arguments starting from a Hamiltonian with quantised field. Here we report a calculation exact in intermolecular correlation up to third order. Subtle optical screening demands a searching logical analysis.

- A-S F Obada and R K Bullough, Optical propagators and properties of the finite molecular crystal,
*Physica***42**(3) (1969), 475-481.We have obtained expressions for the refractive index, rotary power, and binding energy of a molecular crystal of arbitrary symmetry starting from a Hamiltonian with quantised field. We report on the refractive index and assess contributions of the radiation field.

- A-S F Obada and R K Bullough, Dielectric constants for the cubic molecular crystal,
*Chemical Physical Letters***3**(1969), 114-117.The dipole response of a cubic molecular crystal has two parts: the 'virtual' response is not excited by light; the 'real' response is, and is surface dependent. The virtual response is described by essentially two dielectric constants.

- A-S F Obada and R K Bullough, The binding energy of molecular crystals,
*Chemical Physical Letters***3**(1969), 177.We report a dielectric constant formulation of the long wave-length binding energy of a cubic molecular crystal which takes proper account of the radiation field. The binding energy in the p.d.a. is completely determined by the two dispersion relations.

- A-S F Obada and R K Bullough, Longitudinal modes and optical rotation in the finite molecular crystal,
*Physica***42**(1969), 482-488.We report transverse, longitudal and rotatory dispersion relations for a finite molecular crystal of arbitrary symmetry. Generalised surface transmission coefficients are induced by the radiation field. Inclusion of the radiation field does not break the sum-rules.

- R K Bullough, P J Caudrey and A-S F Obada, Theory of radiation reaction and atom self-energies: all-order perturbation theory of the generalized non-relativistic Lamb shift,
*Journal of Physics A: Mathematical, Nuclear and General***7**(13) (1974), 1647-1663.A natural all-order decorrelation procedure is described for the interaction of a single two-level atom with all modes of the radiation field which replaces the semi-classical boson approximation for the atom by an exact fermion treatment. In consequence the vacuum Lamb shift appears as twice the Bethe shift for each level separately instead of the limiting neo-classical shift obtained in the Bose theory. Stimulated emission and absorption modify the Einstein A-coefficient and the vacuum shift is extended to include a field dependent shift. The all-order decorrelated perturbation theory yields an equation of motion for the single-atom dipole moment which includes those reported from operator radiation reaction theory recently; but it also generalizes these by including the effects of stimulated processes. The method is extended to a many-atom case in which it becomes necessary to modify the procedure slightly.

- A-S F Obada, A generalized formula for the binding energy,
*Proceedings of the Seventh Arab Scientific Conference, Cairo 1973*(1974), 173-. - A-S F Obada, A solution to a set of coupled equations for the optical propagators,
*Proceedings of the Seventh Arab Scientific Conference, Cairo 1973*(1974), 119-. - A-S F Obada, On the response of a system in a field,
*Bull. Faculty of Science King Abdulaziz University***1**(1977), 101. - A-S F Obada, M A Allam and A R A El-Samman, Contributions of the correlation functions to the optical constants,
*Proc. Math. Phys. Soc. Egypt No.***46**(1978), 35-42.For a real crystal, the authors define a third-order particle correlation function following the formalism of Van Hove. The correlation function is computed under the Gaussian approximation. In the problem of propagation and scattering of light in crystals, the third-order correlation function already appears in the lowest order terms of the multiple scattering series. The complex refractive index is calculated in the case where multiple scattering is accounted for to second order, which contains third-order correlations. The result is expressed in terms of constants involving rapidly converging lattice sums.

- A-S F Obada and A M Abu-Sitta, Electric- and magnetic-dipole contributions to the reaction field theory,
*Physica A: Statistical Mechanics and its Applications***93**(1-2) (1978), 316-326.A theory for the interaction between electromagnetic radiation and a two-level atom in the electric- and magnetic-dipole approximation is presented. The Dyson-like equation satisfied by the propagators are generalized and a coupled set is obtained. This set is solved, and the susceptibilities are calculated. The dynamical equations for the fields are discussed when the field is supposed to be an unpolarized and isotropic field. It is remarked that the Lamb-shift is enlarged, and the line shape is enhanced. Near resonance the fields are comprised of resonant and off resonant parts. The reaction field is identified.

- A-S F Obada, Response and normal modes of a system in a field,
*Proc. Math. Phys. Soc. Egypt No.***45**(1978), 19-. - A B Khalil and A-S F Obada, Resonances in the electro-magnetic interactions,
*Proc. Math. Phys. Soc. Egypt No.***46**(1978), 29-. - A-S F Obada, A M Abu-Sitta and M H A Mahran, Commutation relations and the moments effects in the theory of quantum optics,
*Proc. Math. Phys. Soc. Egypt No.***47**(1979), 33-40.We describe a natural decorrelation procedure to all orders in the interaction of a single two-level atom with all modes of the radiation field in the electric- and magnetic-dipole approximation. It replaces the semi-classical boson approximation for the atom by an exact fermion treatment. As a consequence, the new polarizabilities are calculated in the two cases of the application of boson and fermion statistics. It is remarked that the energy level splits into a number of peaks. They are shifted and the dependence of line shape on the magnetic moment and occupation number is manifest.

- A-S F Obada, A M M Abu-Sitta and M A El-Deberky, Magnetic-dipole effects in super-radiance,
*Proc. Math. Phys. Soc. Egypt No.***47**(1979), 55-. - A K Abdel-Latif and A-S F Obada, A theoretical investigation of residual stresses,
*Bull. Faculty of Science King Abdulaziz University***3**(1979), 15-. - A-S F Obada and M H Mahran, Response and normal modes of a system in the electric and magnetic-dipole approximation,
*International Centre for Theoretical Physics, Trieste (Italy)*(December 1982), 21 pages.The effect of making the interaction fields evolve from a finite time of a system of molecules is studied in the electric- and magnetic-dipole approximation. Features due to the breaking up of the infinite evolution time are discussed. The resulting equations are analysed for the response of the system. The analysis of the equations for the propagation of normal modes is manifested. The conditions under which normal modes propagate are stated.

- A-S F Obada and M H Mahran, Electric and magnetic dipole contributions to a theory of radiation reaction field and atom self-energy: An operator reaction field,
*Physica A: Statistical Mechanics and its Applications***120**(1-2) (1983), 314-326.The consequences of including magnetic dipole contributions, besides the electric dipole, are considered in the operators for the radiation field. The Bloch equations which describe the two-level atom operators are modified. These equations together with the field operators are discussed, and the contributions are manifested. The spectrum for spontaneous emission and a generalized dynamical Stark effect are obtained. The Rabi frequency is modified.

- A-S F Obada and A B Mayhoup, On the diffusion of pollutants in the atmospheric Ekman boundary layer,
*International Centre for Theoretical Physics, Trieste (Italy)*(October 1983), 6 pages.A model for the pollutants diffusion in the lowest kilometres of the atmosphere, the planetary boundary layer, has been established. In this boundary layer, the vertical gradients of the physical quantities are much larger than the horizontal gradients and the viscosity is important and comparable in magnitude to the pressure gradient and Coriolis forces. The velocity profiles in the layer are derived from Ekman layer equations. The partial differential equation of second order that governing the transport and diffusion of contaminants emitted from an area source has been solved.

- A-S F Obada and M M A Ahmad, Application of the generalized boson operators to a boson-fermion system,
*Optica Acta: International Journal of Optics***32**(8) (1985), 901-906.The generalized boson-fermion interacting system is described by transforming the angular momentum operators into boson operators. Using the Holstein-Primakoff model, the forms for the expectation values of the different number operators are obtained. The second-order correlation function is calculated in different state representations.

- A-S F Obada, A M Abu-Sitta and F K Faramawy, On the generalized linear response functions,
*Physica A: Statistical Mechanics and its Applications***130**(1-2) (1985), 292-322.To discuss the response of a system to an electromagnetic field the minimal coupling Hamiltonian is assumed. The quadratic term in the potential is kept. The Kubo formalism for perturbation is applied and perturbations to all orders in the internal fields are considered. However only the first-order perturbation for the external probe is taken. Thus the linear response function is calculated. The inclusion of the quadratic term contributes terms to this function. Due to these terms the Dyson-like propagator equation is corrected. This equation is developed to obtain the different susceptibilities. It is found that the contributions from the density-current and current-current propagators must be taken into account, in addition to the density-density propagator, when the polarisability is considered to obtain the correct result. The polarisability of the system is then calculated in two approximations. When the quadrupole contribution is considered, the number of propagators increases. They are given through a coupled set of generalized Dyson-like equations. The different fields are defined and discussed and the interplay between them is manifested. The simplest coupled set is considered and developed to give two coupled integral equations in an extension to the classical integral equation for the electric polarization field. They are solved for a finite crystal and the refractive index is calculated for the propagating waves when the probing field is light. The phenomena of birefringence and optical rotation are discussed through simplified models. The extinction theorem is found to hold in a generalized form. Hence the dependence on the geometry of the system is discussed. The effects of the quadrupole terms on these phenomena are shown and discussed. Some other optical phenomena and higher multipole contributions are briefly discussed.

- A-S F Obada, M H Mahran and Z E Mohaned, nlm → n'l'm' transition polarizability, magnetizability and susceptibilities for H atom,
*International Centre for Theoretical Physics, Trieste (Italy)*(February 1985), 13 pages.A closed analytical form of the polarisability, magnetisability and susceptibilities for nlm → n'l'm' transitions is given. As a special case we obtain the earlier 1s → 1s and 1s → 2P

_{z}of Babiker et al. - A-S F Obada and A M Abdel-Hafez, Generalized photon and atomic statistics in Jaynes-Cummings models,
*International Centre for Theoretical Physics, Trieste (Italy)*(August 1985), 35 pages.The interaction between a single two-level and a single mode is studied in some Jaynes-Cummings models. The coupling is supposed to depend on a function of the photon number operator. The evolution operator is calculated; then the probability distribution is obtained. Different statistical quantities are computed for the two states of the atom and different initial distributions for the radiation mode. Three modes are taken, namely, thermal, coherent and squeezed coherent light.

- A-S F Obada, A M M Abu-Sitta and F K Faramawy, Superfluorescence from a system of
*N*-2-level atoms,*First International Conference of Applied Mathematics, 21-26 April 1984, Ain Shams University, Cairo*(1985). - S S Hassan, A-S F Obada and M H Mahran, Magnetic dipole contribution to optical bistability,
*First International Conference of Applied Mathematics, 21-26 April 1984, Ain Shams University, Cairo*(1985). - M Sebawe Abdalla, S S Hassan and A-S F Obada, Multiphoton transition in the Tavis-Cummings model,
*Physical Review A***34**(1986), 4869-4874.The Holstein-Primakoff transformation is used to transform the model of N (>1) two-level atoms collectively interacting with multiphotons of a single quantized mode of the field (the Tavis-Cummings model) into a boson-boson model. Formal solutions are obtained in the strong-coupling limit and/or the large-number-of-atoms limit. The analogy is drawn with a simple model of frequency conversion. The characteristic function in the p representation for the transformed model is calculated.

- A-S F Obada and A M Abdel-Hafez, Photon and atomic statistics in generalized Jaynes-Cummings models,
*Physica A: Statistical Mechanics and its Applications***139**(2-3) (1986), 593-618.The interaction between a single two-level atom and a single mode is studied in some Jaynes-Cummings models. The coupling is supposed to depend on a function of the photon number operator. The evolution operator is calculated; then the probability distribution is obtained. Different statistical quantities are computed for the two states of the atom and different initial distributions for the radiation mode. Three modes are taken and studied, namely thermal, coherent and squeezed coherent light.

- A M Abdel-Hafez, A-S F Obada and M M A Ahmad, Three-level atom and two modes: Generalized models and multiphotons,
*International Centre for Theoretical Physics, Trieste (Italy)*(July 1986), 25 pages.A generalized model is presented for the interaction between a 3-level atom and a 2-mode field. The interaction depends on functions on the photon numbers in the modes. There is also detuning included. The use of spin-1 operators to describe the atomic system is emphasized. Different statistical quantities are computed for the photons and the atomic system in the general case. Multiphoton processes, whether cascade or scattering, are discussed by using specific formulae for the photon numbers functions. A model is presented in which infinite sums tend to elementary functions or infinite integrals in the two cases of thermal and coherent distributions for the two modes.

- A M Abdel-Hafez and A-S F Obada,
*N*-level atom and (*N*-1) modes: Detuning, dynamics and statistical aspects,*International Centre for Theoretical Physics, Trieste (Italy)*(July 1986), 22 pages.A model is presented to investigate the problem of interaction between an

*N*-level atom and (*N*-1) modes of the field. The model includes the effect of detuning. Constants of motion are obtained. The evolution operator is calculated. The probability distribution function for the photons numbers is computed for different initial atomic states. The characteristic functions are computed. Different statistical quantities concerning the photons or the atomic system are given. It is found that if the atom starts from a definite state then the photon number never exceeds its initial value in the mode that effects the interaction between this state and the upper-most state. On the other hand, in the other modes the photon numbers at any time are never less than the initial value. - A-S F Obada, M M A Ahmad and F K Faramawy, Existence of Weisskopf-Wigner approximation for bound-continuous spectral transitions,
*Second International Conference in Applied Mathematics, 23-27 April 1985, Ain Shams University, Cairo*(1986). - A M Abdel-Hafez, A-S F Obada and M M A Ahmad,
*N*-level atom and*N*-1 modes: Statistical aspects and interaction with squeezed light,*Physical Review A***35**(1987), 1634-1647.A model is presented to investigate the problem of interaction between an

*N*-level atom and*N*-1 modes of the field. The model includes detuning. Constants of motion are obtained. The evolution operator is calculated, and the probability distribution function for the photon numbers is computed for different initial atomic states. The characteristic functions are computed. Different statistical quantities concerning the photons or the atomic system are given. The case of a three-level atom and two modes is considered for its different configurations. The phenomenon of collapses and revivals is discussed for squeezed light, and the effect of squeezing is shown in this phenomenon. - A M Abdel-Hafez, A-S F Obada and M M A Ahmad,
*N*-level atom and (N-1) modes: an exactly solvable model with detuning and multiphotons,*J. Phys. A***20**(6) (1987), L359-L364.A model is presented for the interaction between an

*N*-level atom and (*N*-1) modes. For exact resonance between the modes with a detuning parameter, the Hamiltonian becomes solvable. The evolution operator is calculated and then the time development for the photon number operators and the occupation numbers are obtained. Multiphoton processes are discussed. - A M Abdel-Hafez, A-S F Obada and M M A Ahmad,
*N*-level atom and (*N*-1) modes generalized models and multiphotons,*Physica A: Statistical Mechanics and its Applications***144**(2-3) (1987), 530-560.Generalized models are presented for the interaction between an

*N*-level atom and (*N*-1) modes. Spin-1 operators are used to describe the 3-level atom, and the generators of the*U*(*N*) group are used in the general case. Different statistical quantities related to the photons and the atomic system are calculated. Multiphoton processes are discussed. A model is presented in the 3-level atom and 2-mode system, in which infinite series tend to closed forms, for thermal and coherent distributions for the modes. - A-S F Obada and A M Abdel-Hafez, Time Evolution for a Three-level Atom in Interaction with Two Modes,
*Journal of Modern Optics***34**(5) (1987), 665-677.The problem of a three-level atom and two modes is treated quantum mechanically and constants of motion are obtained. The evolution operator is calculated for the case of exact two-photon resonance. The probability distribution functions are calculated for the atom in one of its states. Some statistical quantities of the fields and the atomic systems are given. The interaction with squeezed light is investigated.

- M H Mahran and A-S F Obada, Bunching and Antibunching Effects in a System of N Two-level Atoms,
*Journal of Modern Optics***34**(11) (1987), 1471-1479.The problem of exhibiting bunching and antibunching phenomena for a system of N two-level atoms in interaction with light is investigated. The conditions for the change from bunching to antibunching and vice versa are obtained for resonance, and for time-independent and time-dependent coupling constant. The conditions for the case of resonance and time-independent coupling constant are investigated for both generalized coherent states and squeezed coherent states.

- A-S F Obada, S S Hassan and M H Mahran, A non-linear response theory: I. General formulation,
*Physica A: Statistical Mechanics and its Applications***142**(1-3) (1987), 601-618.We present a generalisation of response theory to the non-linear regime of the interaction of an electromagnetic field with an extended system of two-level atoms. The different propagators together with the equations for the fields with their interdependence are presented. The optical extinction theorem is further generalised. The formulated theory is utilised later to derive the dispersion relations relevant to the non-linear effect of optical multistability.

- S S Hassan, M H Mahran and A-S F Obada, A non-linear response theory: II. Generalised dispersion relations,
*Physica A: Statistical Mechanics and its Applications***142**(1-3) (1987), 619-636.The non-linear response function approach is used to discuss the relationship in the non-linear regime between the different polarisation fields and the impinging field. The dispersion relation is obtained in a generalised form. The relationship between the impinging field and the transmitted field is derived. These relations generalise earlier results obtained in the electric dipole approximation.

- M H Mahran and A-S F Obada, The Dynamics of the Jaynes-Cummings Model in the Logarithmic State,
*Journal of Modern Optics***35**(11) (1988), 1847-1855.The dynamics of the Jaynes-Cummings model (JCM) when the mode initially has a logarithmic distribution is studied. The temporal behaviour of the atomic inversion and the dipole moment are shown. The phenomena of collapses and revivals, bunching, antibunching and squeezing are considered.

- M H Mahran and A-S F Obada, Amplitude-squared squeezing of the Jaynes-Cummings model,
*Physical Review A***40**(1989), 4476-4480.We have investigated the Jaynes-Cummings model for amplitude-squared squeezing. It exists when the atom is initially either in its excited or ground state. It is noted that it occurs during the collapse period.

- A M Abdel-Hafez, A M M Abu-Sitta and A-S F Obada, A generalized Jaynes-Cummings model for the N-level atom and (N-1) modes,
*Phys. A***156**(2) (1989), 689-712.A generalized Jaynes-Cummings model is presented to discuss the interaction between an N-level atom and (N - 1) modes of the radiation field. The level to which the rest of the levels are connected lies in the middle. A detuning parameter is introduced to make the model solvable. Distribution and characteristic functions and different statistical averages are calculated. Multiphoton processes are discussed. The system of a three-level atom and two modes with its three different configurations, and a configuration of the system of a five-level atom and four modes are considered under specified initial atomic conditions. The interaction with squeezed modes of light is investigated. The dependence of the features of the phenomenon of collapses and revivals on the configuration of the system, the atomic initial conditions, the detuning parameter and the statistical aspects of the interacting squeezed modes are shown.

- M H Mahran, A-R A Samman and A-S F Obada, Quantum effects in the three-level two-mode system,
*Journal of Modern Optics***36**(1) (1989), 53-65.The temporal behaviour of the V configuration of a three-level two-mode system is investigated with the two modes initially in a coherent state. The photon number, atomic inversion, reduced atomic fluctuations, squeezing in the field quadratures, cross-correlation between the photon numbers in the two modes, and finally bunching and antibunching effects are investigated. A relation between squeezing and cross-correlation is established.

- M H Mahran and A-S F Obada, Atomic-level shift in a generalized squeezed coherent state,
*Physical Review A***40**(1989), 2215-2216.The reaction field theory is used to calculate the frequency shift in the generalized squeezed coherent state. The result for the squeezed vacuum state is a special case.

- M Sebawe Abdalla, M M A Ahmed and A-S F Obada, Dynamics of a non-linear Jaynes-Cummings model,
*Physica A: Statistical Mechanics and its Applications***162**(2) (1990), 215-240.A non-linear Jaynes-Cummings model is presented. The atom interacts with two modes non-linearly. One- and multiphoton processes are considered. The Heisenberg equations of motion are solved exactly with detuning present. The evolution operator is calculated for the general case. Different statistical quantities pertaining to either the atom or the fields are computed. Numerical investigations are carried out for the phenomena of collapses and revivals, bunching and antibunching, modes correlation and squeezing for different initial values of the mean number of photons.

- M H Mahran and A-S F Obada, Amplitude-squared squeezing in the multiphoton Jaynes-Cummings model,
*Physical Review A***42**(1990), 1718-1724.Amplitude-squared squeezing has been investigated for the generalized multiphoton Jaynes-Cummings model. Different values for the initial photon numbers as well as different photon multiplicities have been used. The atom is taken to be in either the ground or the excited state. The relation between both amplitude-squared and normal squeezing and the phenomenon of collapse and revival has been discussed.

- M Sebawe Abdalla, M M A Ahmed and A-S F Obada, Multimode and multiphoton processes in a non-linear Jaynes-Cummings model,
*Physica A: Statistical Mechanics and its Applications***170**(2) (1991), 393-414.Exact operator solutions for the generalized Jaynes-Cummings model (JCM) of a two-level atom interacting with a non-linear bosonic field with multimodes has been given. Two different methods have been applied, the Heisenberg equations and the evolution operator. The photon numbers have been given through the characteristic function. Different statistical quantities have been computed. The phenomenon of collapses and revivals in the atomic inversion and the photon number as well as squeezing have been investigated.

- A-S F Obada and A M Abdel-Hafez, Effect of atomic coherence on squeezing,
*Physical Review A***43**(1991), 5161-5164.The dependence of the phenomenon of squeezing for the field quadratures on the phases of the field and atomic coherent states when the atom is assumed to start in an atomic coherent state is investigated. It is found that increasing the relative phase difference results in larger amounts of squeezing for fixed

*θ*. An explanation for the occurrence of squeezing is given on the idea of the interference between the slowly varying and fast oscillating terms. - A M Abdel-Hafez and A-S F Obada, Amplitude-squared squeezing in the multiphoton Jaynes-Cummings model: Effect of phases,
*Physical Review A***44**(1991), 6017-6022.The phenomenon of amplitude-squared squeezing is investigated for the one- and two-photon Jaynes-Cummings model. The relative phase between the atomic coherent state and the coherent field state is found to affect this phenomenon with larger amounts of squeezing that can be found by changing the angles

*θ*for fixed relative phase. - F K Faramawy, M M Abu Sitta and A-S F Obada, Propagation in non-linear media: Refractive index in non-linear media,
*International Centre for Theoretical Physics, Trieste (Italy)*(August 1991), 22 pages.A general theory of the effect of the electronic coupling between molecules on the response of the system as a whole to an external electromagnetic field is presented. The role of the non-linear response function for various physical quantities and their Fourier transformation is discussed. The non-linear susceptibilities to any rank for a denaturated system of particles are calculated. Also the non-linear response function for a coupled system is computed. A non-linear susceptibility for the electric dipole is obtained. Methods of the linear theory are developed to obtain the total non-linear refractive index.

- F K Faramawy, M M Abu Sitta and A-S F Obada, Propagation in non-linear media: Refractive index in non-linear media,
*LAMP Series Report/91/7**International Centre for Theoretical Physics, Trieste (Italy)*(August 1991), 22 pages.A general theory of the effect of the electronic coupling between molecules on the response of the system as a whole to an external electromagnetic field is presented. The role of the non-linear response function for various physical quantities and their Fourier transformation is discussed. The non-linear susceptibilities to any rank for a denaturated system of particles are calculated. Also the non-linear response function for a coupled system is computed. A non-linear susceptibility for the electric dipole is obtained. Methods of the linear theory are developed to obtain the total non-linear refractive index.

- A-S F Obada, Non-classical effects in some JC models,
*First International Conference of Mathematics and Statistics Assuit, Egypt 12-14 February 1990***4**(1991), 59-. - A M Abdel-Hafez and A-S F Obada, Non-classical effects in Jaynes-Cummings model: effect of the phases,
*Physica A: Statistical Mechanics and its Applications***180**(1-2) (1992), 205-224.The effect of the change of phases in the field coherent state and in the atomic coherent state on the raduction of the fluctuations in atomic variables, and on the normal and amplitude-squared squeezing in the Jaynes-Cummings model is investigated. It is found that these phenomena are sensitive to the changes of phase in the atomic and field coherent states. The occurrence of the amplitude-squared squeezing during the period of collapse is shown to be the result of interference between the slowly oscillating functions in the expressions for the parameters of these phenomena.

- A-S F Obada and A M Abdel-Hafez, Amplitude-squared and frequency sum squeezing in the three-level atom,
*Optics Communications***91**(1-2) (1992), 99-103.The phenomena of amplitude-squared and frequency sum squeezing are investigated for a three-level atom and two modes. The effects of detuning on these phenomena are studied for one-photon and two-photon interactions. The relation between the collapse and revival phenomena and amplitude-squared and frequency sum squeezing is shown. For off-resonance it is found that the frequency sum squeezing is improved more than amplitude-squared squeezing for the one-photon interaction. While for the two- photon interaction it is the other way around.

- A M Abdel-Hafez and A-S F Obada, The amplitude-squared, sum and difference squeezing in the interaction of two quantized field modes with a two-level atom,
*Physica Scripta***46**(1992), 27-30.The sum, difference and amplitude-squared squeezing are investigated for a generalized Jaynes-Cummings model with two field modes. Different initial states for the two-level atom are considered, and the dependence on the relative phases between the atomic and field coherent states is investigated.

- A M Abdel-Hafez and A-S F Obada, Effect of detuning and coherence phase on normal squeezing,
*Annales de Physique***17**(4) (1992), 337-348.The effects of detuning as well as the relative phases between the atomic coherence and the field coherent state on the squeezing phenomenon for the one-photon and two-photon Jaynes-Cummings model are investigated. Amounts of normal squeezing can be improved by appropriate choices of detuning and phases. The relation between the collapses and revivals phenomenon and squeezing is made transparent.

- S S Hassan, M Sebawe Abdalla, A-S F Obada and H A Batarfi, Periodic squeezing in the Tavis-Cummings model,
*J. Modern Opt.***40**(7) (1993), 1351-1367.By utilizing our previous operator solution, we have investigated the squeezing in the radiation field of the Tavis-Cummings model (collective N >> 1 two-level atoms interacting with a resonant single cavity quantized mode). With field and atoms initially in coherent field state strong or weak and atomic coherent state (of few excited atoms), periodic time-dependent squeezing in the field and the macroscopic polarization is expressed in terms of Jacobian elliptic functions of the first kind. The statistical investigations are carried out for the quasi-probability distribution functions (Wigner function and Q function). The distribution function of the field quadrature has a variance less (greater) than that for a coherent state if this quadrature is squeezed (unsqueezed).

- A-S F Obada, S S Hassan, R R Puri and M Sebawe Abdalla, Variation from number- to chaotic-state fields: A generalized geometric state,
*Physical Review A***48**(1993), 3174-3185.We introduce a formal state of the radiation field which interpolates between the (pure) number (Fock) state and the (nonpure) chaotic state. The correlation functions and the squeezing properties are studied. The study of two quantum optical systems (namely, the Jaynes-Cummings model and resonance fluorescence for a single atom and many cooperative atoms) in such a state of the radiation field provides more insight into the gradual behaviour as one goes from the number state to the chaotic state. A scheme for the production of such a state is sought to be realized in a model of multiphoton processes in a finite-level atomic system. The quasi-probability Wigner distribution function for such a state is also examined.

- A-S F Obada and Z M Omar, Non-classical effects in two two-level atoms and a single mode,
*J. Egypt Math. Soc.***1**(1993), 63-. - M Sebawe Abdalla, M H Mahran and A-S F Obada, Statistical properties of the even binomial state,
*J. Modern Opt.***41**(10) (1994), 1889-1902.The even binomial states are considered and their properties are studied. It is shown that they interpolate between the even number eigenstates and the even coherent states. Their Q-representation and their Wigner function are calculated and their squeezing properties are discussed.

- A-S F Obada, A M M Abu-Sitta and O M Yassin, Three-Photon Micromaser,
*J. Egypt Math. Soc.***2**(1994), 103-. - H A Batarfi, M Sebawe Abdalla, A-S F Obada and S S Hassan, Field distribution in a generalized geometric radiation state,
*Physical Review A***51**(1995), 2644-2647.Both the Wigner function and the field distribution function in the generalized geometric state of the radiation field, which were recently introduced by us [A-S F Obada, S S Hassan, R R Puri, and M S Abdalla,

*Phys. Rev. A***48**, 3174 (1993)], are calculated exactly and examined in both the phase-averaged and the nonaveraged cases. - A M Abdel-Hafez, A-S F Obada and A H Essawy, Phase Properties of Multiphoton Jaynes-Cummings Model,
*Annales de Physique***20**(1) (1995), 47-61.The multiphoton Jaynes-Cummings model is analysed for phase properties of a coherent field state. The phase variances tend to the values for a field with a randomly distributed phase of one-photon Jaynes-Cummings model and the off-resonant two-photon Jaynes-Cummings model. Revival of oscillations in the far off-resonant two-photon Jaynes-Cummings model occur much later. On the other hand, the variance of the on-resonant two-photon Jaynes-Cummings model variances oscillate periodically.

- A-S F Obada and Z M Omar, Dynamics of two three-level atoms interacting with two modes of radiation,
*LAMP Series Report/95/9, International Centre for Theoretical Physics, Trieste (Italy)*(December 1995), 14 pages.The system of two 3-level atoms in interaction with 2-modes of the radiation field is investigated. The (H cascade) configuration for the two identical atoms is taken and exact resonance is considered. The wave function is calculated and the phenomena of collapses and revivals, sub-poissonian statistics, and two-mode anti-correlation are discussed. Subrevivals are noted for these phenomena.

- A-S F Obada and Z M Omar, Properties of squeezed Schrödinger cats,
*International Centre for Theoretical Physics, Trieste (Italy)*(September 1995), 19 pages.In this article we investigate some statistical properties of the even and odd squeezed (squeezed Schrödinger cat) states. The quasi-probability distribution functions especially W(±) and Q(±) are calculated and discussed for these states. The phase distribution function is discussed. A generation scheme is proposed for either the squeezed generalized Schrödinger cat, or the squeezed number state.

- A-S F Obada and Z M Omar, Dynamics of two three-level atoms interacting with two modes of radiation,
*International Centre for Theoretical Physics, Trieste (Italy)*(December 1995), 14 pages.The system of two 3-level atoms in interaction with 2-modes of the radiation field is investigated. The ( cascade) configuration for the two identical atoms is taken and exact resonance is considered. The wave function is calculated and the phenomena of collapses and revivals, sub-poissonian statistics, and two-mode anti-correlation are discussed. Subrevivals are noted for these phenomena.

- A-S F Obada, M H Mahran, F A A El-Orany and M Sebawe Abdalla, Dynamics of the even-binomial state in some quantum system,
*International Journal of Theoretical Physics***35**(7) (1996), 1393-1404.The effect of the even-binomial state (the state which interpolates between the even-number state and the even-coherent state) on the Jaynes-Cummings model and resonance fluorescence for a single atom and for many cooperative atoms is discussed. The study of two quantum optical systems in such a state provides more insight into the continuous passage from the even-number state to the even-coherent state. The squeezing phenomenon and the correlation functions are examined.

- M H Mahran and A-S F Obada, Dynamics of generalized Jaynes-Cummings model in the binomial state,
*Proceedings of First International Science Conference (Science & Development), Faculty of Science, Al-Azhar University, Cairo 20-23 March 1995*(1996), 403-410. - A-S F Obada, M H Mahran and M A El-Deberky, Detuning effects on a nonlinear Jaynes-Cummings model,
*Proceedings of First International Science Conference (Science & Development), Faculty of Science, Al-Azhar University, Cairo 20-23 March 1995*(1996), 433-440. - A-S F Obada and Z M Omar, Properties of superposition of squeezed states,
*Physics Letters A***227**(5-6) ( 1997), 349-356.We investigate some statistical properties of superposed squeezed states. The quasi-probability distribution functions, especially W(±) and Q(±), are calculated and discussed for these states. A generation scheme is proposed for both the squeezed generalized Schrödinger cat, or the squeezed number state.

- A-S F Obada, O M Yassin and S M Barnett, Phase properties of coherent phase and generalized geometric states,
*Journal of Modern Optics***44**(1) (1997), 149-161.We examine some properties of a class of partial phase states of a single field mode. The quasi-probability distribution function and the phase properties of both the generalized geometric and even geometric states are investigated. The relation between the coherent phase states and the SU(1, 1) coherent states is sought.

- A Joshi and A-S F Obada, Some statistical properties of the even and the odd negative binomial states,
*J. Phys. A***30**(1) (1997), 81-97.New states of an electromagnetic field, even and odd negative binomial states, are introduced here. These states interpolate between the even [odd] coherent states and the even [odd] quasi-thermal states. Various statistical properties of these states such as the mean photon number, second-order coherence function and normal squeezing are calculated. The Wigner functions for these states are also discussed. These results may be useful for carrying out a systematic study of the non-classical properties of 'Schrödinger cat states' as one moves from an even [odd] coherent state to an even [odd] quasi-thermal state.

- A-S F Obada and G M Abd Al-Kader, S-parameterised quasi-probability distribution functions on a linear amplifier,
*Journal of Physics B: Atomic, Molecular and Optical Physics***30**(22) (1997), 5109-5121.The s-parameterised characteristic function for the output light is given. The s-ordered distribution functions for the output of a linear amplifier are investigated. The input light field of the linear amplifier is assumed to be a squeezed displaced Fock state. Various moments are calculated by using the s-ordered characteristic function for the output linear amplifier. The various distribution functions of the output fields are plotted as functions of the interaction time. The phase distribution for the output light is discussed.

- A-S F Obada, A M Abdel-Hafez and M Abdelaty, Phase properties of a Jaynes-Cummings model with Stark shift and Kerr medium,
*The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics***3**(3) (1998), 289-294.Phase properties of the field interacting with a two-level atom in a lossless cavity Jaynes-Cummings model, taking into account the level shifts produced by Stark effect with an additional Kerr medium for one-mode are studied using the phase formalism of Pegg and Barnett. It is shown in particular that phase properties of the field reflect the collapse and revival phenomena. The results for the time evolution of the phase probability distribution and the phase fluctuations are obtained. The effect of Stark shift on the phase properties in both the absence and presence of a Kerr medium is analysed. Phase localization is found for certain choice of the parameters.

- A-S F Obada and G M Abd Al-Kader, Squeezed displaced Fock states and non-diagonal P representation,
*Journal of Modern Optics***45**(4) (1998), 713-734.An R representation with a squeezed displaced-Fock-states (SDFSs) basis is introduced. A non-diagonal P representation for the density operator with SDFS basis is defined. The special cases for the fields, chaotic and laser, are calculated. The Pegg-Barnett phase distribution is given and compared with the radial integration of the generalized P function. The Fokker-Planck equation for the damped harmonic oscillator is considered, and its steady-state solution is given.

- A-S F Obada, A M Abdel-Hafez and H A Hessian, Influence of intrinsic decoherence on non-classical effects in the non-degenerate bimodal multiquanta Jaynes-Cummings model,
*The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics***31**(23) (1998), 5085-5104.The non-degenerate bimodal multiquanta Jaynes-Cummings model (JCM) governed by the Milburn equation is studied. An exact solution of this equation is obtained and applied to investigate the influence of the intrinsic decoherence on non-classical effects (population inversion, oscillations of the number distribution, squeezing of the field modes), for the resonant and the off-resonant cases.

- A-S F Obada and G M Abd Al-Kader, Superposition of displaced Fock states: properties and generation,
*Acta Physica Slovaca***48**(5) (1998), 583-594.The superposition of displaced Fock states is discussed and various moments are calculated. The s-ordered quasi probability distribution function for the superposition of displaced Fock states is investigated. The s-parameterized characteristic function for the superposition of two displaced Fock states are considered. The Glauber second-order coherence function is calculated. The squeezing properties of the superposition of displaced Fock states are studied. Analytical and numerical results for the quadrature component distributions are presented. A generation scheme is discussed.

- A-S F Obada and G M Abd Al-Kader, Squeezed states and non-diagonal P-representation,
*Internat. J. Theoret. Phys.***37**(8) (1998), 2207-2232.The R-representation and the non-diagonal P-representation for density operator with squeezed-state basis are defined. The special cases for chaotic and laser fields are calculated. The Fokker-Planck equation for the damped harmonic oscillator with squeezed bath is considered and the steady-state solution is given. A special case of the steady-state solution for only a thermal bath is shown. The Pegg-Barnett phase distribution is compared with the radial integration on the generalized P-function.

- A-S F Obada and G M Abd Al-Kader, Linear amplifier and quasi-probability distribution functions for the squeezed displaced Fock states,
*Internat. J. Theoret. Phys.***37**(8) (1998), 2233-2260.The non-diagonal P-function (complex and positive) for the output field with the squeezed coherent state as an initial state to a linear amplifier is obtained. Moments of the field operators are calculated. The Wigner distribution function for the output of a linear amplifier is discussed. The input light field of the linear amplifier is assumed to be a squeezed displaced Fock state. Investigation of the Wigner distribution functions at the output fields is carried out, where it is found that the distribution spreads out as it rotates in the phase space with time development. For the case of a squeezed displaced Fock state input it is found that the distribution disintegrates into two subdistributions. The phase distribution for the output light is discussed through the phase function associated with the Wigner function for different inputs.

- A-S F Obada and G M Abd Al-Kader, S-parameterized quasi-probability for squeezed displaced Fock states,
*Nonlinear Optics***19**(1998), 143-156. - F A A El-Orany, M H Mahran and A-S F Obada, Statistical properties and dynamical behaviour in odd binomial states,
*Acta Univ. Palacki. Olomuc. Fac. Rer. Nat. Physica***37**(1998), 39-60. - A-S F Obada and G M Abd Al-Kader, Superpositions of squeezed displaced Fock states: Properties and generation,
*Journal of Modern Optics***46**(2) (1999), 263-278.The s-parametrised characteristic function for the superposition of squeezed displaced Fock states (SDFSs) is given. The s-parametrised distribution functions for the superposition of SDFSs are investigated. Various moments are calculated by using this characteristic function. The Glauber second-order coherence function is calculated. The squeezing properties of the superposition of SDFSs are studied. Analytical and numerical results for the quadrature component distributions for the superposition of a pair of SDFSs are presented. A generation scheme is discussed.

- F A A El-Orany, M H Mahran, A-S F Obada and M Sebawe Abdalla, Statistical properties of the odd binomial states with dynamical applications,
*Internat. J. Theoret. Phys.***38**(5) (1999), 1493-1520.We investigate the odd binomial state, which interpolates between odd number and odd coherent states. We discuss the statistical properties of the Glauber second-order correlation function and squeezing phenomena (normal squeezing and amplitude-squared squeezing). The phase properties in Pegg-Barnett formalism are considered, and the quasi-probability distribution functions are examined. The dynamics of the Jaynes-Cummings model and the resonance fluorescence are also discussed.

- A-S F Obada, A M Abdel-Hafez, A H Essawy and M Abdelaty, Phase properties of a field mode interacting with a three-level atom with Kerr nonlinearity,
*Physica Scripta***60**(4) (1999), 329-334.Phase properties of a field interacting with a three-level atom with a Kerr-like medium are investigated through the phase formalism of Pegg and Barnett. The phase probability distribution, the expectation value and the phase fluctuations of the Hermitian phase operator are calculated. The phase properties of the field reflect the collapses and revivals. A number of interesting features are found and discussed.

- A-S F Obada and G M Abd Al-Kader, Superpositions of squeezed coherent states: properties and generation,
*International Journal of Modern Physics B***13**(17) (1999), 2299-2312.The s-parameterized characteristic function for the superposition of squeezed coherent states (SCS's) is given. The s-parameterized distribution functions for the superposition of SCS's are investigated. Various moments are calculated by using this characteristic function. The Glauber second-order coherence function is calculated. The photon number distribution of the superposition of SCS's studied. Analytical and numerical results for the quadrature component distributions for the superposition of a pair of SCS's are presented. The phase distribution calculated from the integration of s-parameterized distribution function over the phase space. A generation scheme is discussed.

- A-S F Obada, A M Abdel-Hafez and M Abdel-Aty, Phase properties of two-mode optical fields with Stark shift and Kerr nonlinearity,
*Journal of the Physical Society of Japan***68**(1999), 2269-2275.The influence of Stark shift on the phase properties when an ideal cavity is filled with a Kerr-like medium and coupling is effected through non-degenerate Raman two-photon processes is studied. This has been carried out using the Hermitian phase formalism of Pegg and Barnett. We find that the phase properties of the field reflect the collapses and revivals of the level occupation probabilities in most situations. The interaction brings about the symmetrical splitting of the phase probability distribution. A number of interesting features are found and discussed.

- A-S F Obada, A M Abdel-Hafez and H A Hessian, Intrinsic decoherence in no-degenerate two-quanta Jaynes-Cummings mode,
*Acta Physica Slovaca***49**(3) (1999), 381-398.The non-degenerate two-quanta Jaynes-Cummings model governed by Milburn equation is studied. This models the decoherence of a quantum system as it evolves through intrinsic mechanisms beyond conventional quantum mechanics governed by the Schrödinger equation. The exact solution is given. Then it is applied to investigate the effects of the intrinsic decoherence on non-classical effects (population inversion, oscillations of the number distribution, photon antibunching effect, the cross correlation and squeezing of the cavity field). We show how the intrinsic decoherence suppresses the non-classical behaviour in this model, for the off-resonant case.

- A-S F Obada and M Abdel-Aty, Influence of the Stark shift and Kerr-like medium on the evolution of field entropy and entanglement in two-photon processes,
*Acta Physica Polonica B***31**(3) (2000), 589-599.We have investigated the evolution of the field quantum entropy and the entanglement of the atom-field in the two-photon process, taking into account the level shifts produced by Stark effect with an additional Kerr-like medium for one mode. The exact results are employed to perform a careful investigation of the temporal evolution of the entropy. A factorization of the initial density operator is assumed, with the privileged field mode being in a coherent state. We invoke the mathematical notion of maximum variation of a function to construct a measure for entropy fluctuations. The effect of both the Stark shift and the presence of a Kerr-like medium on the entropy is analysed. It is shown that the addition of the Kerr medium and the Stark shift has an important effect on the properties of the entropy and entanglement. The results show that, the effect of the Kerr medium and the Stark shift changes the quasi-period of the field entropy evolution and entanglement between the atom and the field. The general conclusions reached are illustrated by numerical results.

- A-S F Obada, A M Abdel-Hafez and H A Hessian, Two-mode normal squeezing of a nondegenerate bimodal multiquanta Jaynes-Cummings model in the presence of Stark shifts,
*International Journal of Theoretical Physics***39**(6) (2000), 1499-1513.We investigate the two-mode normal squeezing in the presence of Stark shifts for a generalized Jaynes-Cummings Model (JCM) with a two-level particle (atom or trapped ion) and two modes interacting nonlinearly. The effect of the relative phase between the atomic superposition state and the coherent field in the presence and absence of the Stark shift on squeezing is studied. Different values for the parameters of the atomic coherent state are taken.

- M Abdel-Aty, S Abdel-Khalek and A-S F Obada, Pancharatnam phase of two-mode optical fields with Kerr nonlinearity,
*Optical Review***7**(6) (2000), 499-504.The Pancharatnam phase when an ideal cavity is filled with a Kerr-like medium and coupling is affected through a non-degenerate Raman two-photon process is studied. A careful investigation is made seeking exact results on the temporal evolution of atomic inversion and Pancharatnam phase. We invoke the mathematical notion of maximum variation of a function to construct a measure for Pancharatnam phase fluctuations. It is shown that Pancharatnam phase contains explicit information about the statistics of the field and atomic coherence, and is also shown that addition of the Kerr medium has an important effect on the properties of this phase. The results show that the effect of the Kerr medium changes the quasi-period of the Pancharatnam phase evolution. The influence of Stark shift on the atomic inversion and the Pancharatnam phase in both presence and absence of the nonlinear medium is examined. General conclusions reached are illustrated by numerical results.

- M Sebawe Abdalla and A-S F Obada, Quantum Statistics of a New Two Mode Squeeze Operator Model,
*Internat. J. Modern Phys. B***14**(10) (2000), 1105-1128.In this paper we introduce a new squeeze operator which is a combination of the two single photon squeezing operator and two mode correlated operator. We construct the general coherent-state wave function, and study the effects of this operator on the photon number sum and difference. We find in absence of the squeezing parameters lð_1 and lð _2 both the sum and the difference are conserved by the two mode correlated operator. The squeezing phenomenon is discussed, and the minimum uncertainty state is found for small value of the squeezing parameter lð. Besides the Glauber second order correlation function, the statistical investigations are carried out for the quasi-probability distribution functions (P-representation, Wigner function and Q-function). The discussion related to the phase space distribution function is given, and the matrix elements of the density operator are calculated. The photocount distribution is also considered.

- F A A El-Orany, M Sebawe Abdalla, A-S F Obada and G M Abd Al-Kader, Influence of squeezing operator on the quantum properties of various binomial states,
*Internat. J. Modern Phys. B***15**(1) (2001), 75-100.In this communication we investigate the action of a single-mode squeeze operator on the statistical behaviour of different binomial states. For the resulting states (squeezed generalized binomial states) normalized second-order correlation function, quasi-probability distribution functions and the distribution function P(x) associated with the quadrature x are studied both analytically and numerically. Furthermore, the quadrature phase distribution as well as the phase distribution in the framework of Pegg-Barnett formalism are discussed.

- M Abdel-Aty, G M Abd Al-Kader and A-S F Obada, Entropy and entanglement of an effective two-level atom interacting with two quantized field modes in squeezed displaced Fock states,
*Chaos Solitons & Fractals***12**(13) (2001), 2455-2470.We study the influences of ac-Stark shifts on the field quantum entropy, with 'squeezed displaced Fock states' (SDFS) basis. By a unitary transformation we derive a Raman-coupled Hamiltonian perturbatively in the coupling constants. The exact results are employed to perform a careful investigation of the temporal evolution of entropy. A factorization of the initial density operator is assumed, with the privileged field mode being in the SDFS. We invoke the mathematical notion of maximum variation of a function to construct a measure for entropy fluctuations. The results show that the effect of the SDFS changes the quasi-period of the field entropy evolution and entanglement between the atom and the field. The Rabi oscillation frequency, the collapse and revival times of the atomic coherence are found to have strikingly different photon-intensity dependent than those found previously. The general conclusions reached are illustrated by numerical results.

- A-S F Obada and G M Abd Al-Kader, Generation and properties of a superposition of four displaced Fock states,
*Internat. J. Theoret. Phys.***40**(10) (2001), 1715-1735.A detailed discussion of a type of four-component superposition of displaced Fock states is presented. A generation scheme is proposed for these states. The s-parameterized characteristic function and the quasi-probability distribution functions of these states are calculated. The non-classical properties of these states such as photon number distribution and squeezing are discussed. The quadrature distributions are illustrated. The Pegg-Barnett phase distribution is discussed.

- M Abdel-Aty, M Sebawe Abdalla and A-S F Obada, Quantum information and entropy squeezing of a two-level atom with a non-linear medium,
*J. Phys. A***34**(43) (2001), 9129-9141.In the language of quantum information theory we study the entropy squeezing of a two-level atom in a Kerr-like medium. A definition of squeezing is presented for this system, based on information theory. The utility of the definition is illustrated by examining the entropy squeezing of a two-level atom with a Kerr-like medium. The influence of the non-linear interaction of the Kerr medium, the atomic coherence and the detuning parameter on the properties of the entropy and squeezing of the atomic variables is examined.

- M Abdel-Aty, S Abdel-Khalek and A-S F Obada, Entropy evolution of the bimodal field interacting with an effective two-level atom via the Raman transition in Kerr medium,
*Chaos, Solitons & Fractals***12**(11) (2001), 2015-2022.For a general class of two-mode, simple analytic expressions are derived for the evolution of the field quantum entropy in the bimodal field interacting with an effective two-level atom via the Raman transition, with an additional Kerr-like medium. The effect of a Kerr-like medium on the entropy is analysed. It is shown that the addition of the Kerr medium has an important effect on the properties of the entropy and the entanglement. The results show that the effect of the Kerr medium changes the quasi-period of the field entropy evolution and entanglement between the atom and the field. The general conclusions reached are illustrated by numerical results.

- A A Ashour and A-S F Obada (eds.),
*Mathematics and the 21st Century: Proceedings of the International Conference, Cairo, Egypt, 15-20 January 2000*(World Scientific, 2001). - M Sebawe Abdalla and A-S F Obada, Intermediate states: some non-classical properties, in
*Mathematics and the 21st century, Cairo, 2000*(World Sci. Publ., River Edge, NJ, 2001), 323-356.In this article we consider in some detail some new classes of states. These states are intermediate states either between the pure number (Fock) states and the (non-pure) chaotic state (thermal state), such as the geometric state, or between the coherent state and a number state such as the binomial state. We extend our discussion to include some other states such as even (odd) coherent states, even (odd) binomial states, phased generalized binomial states, etc. In our study of these states we pay attention to a discussion of the non-classical properties, besides the statistical properties, for example correlation functions, squeezing, and quasi-probability distribution functions (P-representation, W-Wigner, and Q-function). Furthermore, we consider the field distribution and the photon number distribution, as well as the phase properties. Finally, some schemes for the production of these states are presented.

- A-S F Obada, G M Abdal-Kader and M Abdel-Aty, Aspects of the two-level atom in squeezed displaced Fock states,
*Physica Scripta***64**(6) (2001), 573-579.This paper presents some results on aspects of the two-level atom interacting with a single-mode with the privileged field mode being in the squeezed displaced Fock state (SDFS). The exact results are employed to perform a careful investigation of the temporal evolution of the atomic inversion, entropy and phase distribution. It is shown that the interference between component states leads to non-classical oscillations in the photon number distribution. At the revival time the field is again in the pure state. We have briefly discussed the collapses and revivals of the atomic inversion by studying the evolution of the Q function of the cavity field. We also established the connection between the field entropy and the collapses and revivals of the atomic inversion. We find that the phase probability distribution of the field reflects the collapses and revivals of the level occupation probabilities in most situations. The interaction brings about the symmetrical splitting of the phase probability distribution. The general conclusions reached are illustrated by numerical results.

- M Abdel-Aty and A-S F Obada, Semi-classical stability analysis of a two-photon laser including spatial variation of the cavity field,
*The European Physical Journal D - Atomic, Molecular, Optical and Plasma***15**(3) (2001), 385-391.We investigate the dynamics of a two-photon laser under conditions where the spatial variation of the cavity field along the cavity axis is important. The model assumes pumping to the upper state of the two-photon transition. We consider the Maxwell-Bloch equations on the basis of which we study the stability analysis of the steady state of the system. The system is taken to be contained in a ring-laser cavity. Asymptotic expansion of the eigenvalue and analytic information are obtained in some realistic limits, such as very large reflectivity, very small cavity losses, or very small population relaxation rate. The results are illustrated with an application to a specific atomic system (potassium) as an amplifying medium.

- M Abdel-Aty and A-S F Obada, Stability and instability analysis in two-photon laser,
*Proceedings of Sixth International Al-Azhar Engineering Congference, Cairo, 1-4 September 2000***9**(2001), 431-439. - A-S F Obada and M M Nassar, Discrete Wigner function and quantum state tomography,
*Proceedings of Fourth International Science Conference (Science & Development), Faculty of Science, Al-Azhar University, Cairo 27-29 March 2001*(2002). - A-S F Obada and G M Abdal-Kader, Four-component superposition of squeezed displaced Fock states,
*Chaos Solitons & Fractals***13**(2002), 115-130.A detailed discussion of a type of four-component superposition of squeezed displaced Fock states (SDFSs) is presented. These states, often called Schrödinger cat states for the coherent states, are of great interest at present. The s-parameterized characteristic function (CF) and the quasi-probability distribution functions (QDFs) of these states are calculated. The non-classical properties of these states such as photon number distribution and squeezing are discussed. The quadrature distributions are illustrated. The Pegg-Barnet phase distribution is discussed. A generation scheme is proposed for these states.

- M Abdel-Aty, T M El-Shahat and A-S F Obada, Properties of entropy and entanglement of two-mode nonlinear coherent states,
*Commun. Theor. Phys. (Beijing)***38**(6) (2002), 715-728.In this communication, two-mode nonlinear coherent states are reviewed and special cases are given. Starting from a three-level atom coupled to two modes of radiation with any form of nonlinearities of the two-mode fields, we derive a Raman-coupled effective Hamiltonian by a unitary transformation, evaluated perturbatively in coupling constants. We use the quantum entropy to measure the degree of entanglement in the time development of an effective two-level atom interacting with two-mode nonlinear-coherent states. The results show that the nonlinearity effect yields the superstructure of atomic Rabi oscillations and the effect of the Stark shift changes the quasi-period of the field entropy evolution and entanglement between the particle and the field. A possible experimental test of a new effect is proposed.

- A-S F Obada and G M Abdal-Kader, Four-component superposition of squeezed coherent states,
*Int. J. Theor. Phys. Group Theory Nonlinear Opt.***8**(3) (2002), 293-318.A detailed discussion of a type of four-component superposition of squeezed states (SCS's) is presented. These states, which may be called squeezed Schrödinger cat states, are of great interest at present. The s-parameterized characteristic function (CF) of these states is calculated. Some non-classical properties of these states such as photon number distribution, correlation function, and squeezing as well as the Pegg-Barnett phase distribution and the quadrature phase distributions are discussed. The Q and Wigner quasi-probability distribution functions are investigated. We propose a scheme for generating four-component superposition of (SCS's) depending on the motion of a trapped ion. In the scheme the ion is exited by a sequence of laser pulses tuned to the respective vibrational sidebands. An SU(1, 1) Lie group analytic representation of four-component SCS's in unit disk based on the over-complete set of the SU(1, 1) Perelomov coherent states is discussed.

- M Abdel-Aty and A-S F Obada, Quantum inversion of cold atoms in a microcavity: Spatial dependence,
*J. Phys. B: At. Mol. Opt. Phys.***35**(4) (2002), 1-7.The atomic inversion of cold atoms in a microcavity when its centre-of-mass (CM) motion is described quantum mechanically is presented, but is distinguished from other treatments by the inclusion of the spatial variation along the cavity axis. It is shown that the spatial dependence of the atomic inversion is very sensitive to changes of the atomic CM motion, as well as the cavity length. For ultracold atoms, different behaviours in the atomic inversion are demonstrated for the change in cavity length. The general conclusions reached are illustrated by numerical results.

- M Abdel-Aty, S Furuichi and A-S F Obada, Entanglement degree of a nonlinear multiphoton Jaynes-Cummings model,
*J. Opt. B: Quantum Semiclass Opt.***4**(2002), 37-43.We extend our earlier investigations (S Furuichi and M Abdel-Aty,

*J. Phys. A: Math. Gen.***34**( 2001), 6851) on the entanglement degree of a two-level atom to include any forms of nonlinearities of both the field and the intensity-dependent atom-field coupling. We present a derivation of the unitary operator within the frame of the dressed state approach, by means of which we identify and numerically demonstrate the region of parameters where significantly large entanglement can be obtained. The influences of the nonlinearity, Stark shifts and detuning on the degree of entanglement are examined. It is shown that features of the degree of entanglement are influenced significantly by the kinds of nonlinearity of the single-mode field. The model presented in this paper can be regarded as a generalization of the Jaynes-Cummings model. - N A Zidan, M Abdel-Aty and A-S F Obada, Influence of intrinsic decoherence on entanglement degree in the atom-field coupling system,
*Chaos Solitons & Fractals***13**(2002), 1421-1428.In this paper, we use the quantum mutual entropy to measure the degree of entanglement in the time development of a two-level particle (atom or trapped ion). We find an exact solution of the Milburn equation for the system. The exact solution is then used to discuss the influence of intrinsic decoherence on degree of entanglement. The exact results are employed to perform a careful investigation of the temporal evolution of the entropy. It is shown that the degree of entanglement is very sensitive to the changes of the intrinsic decoherence. The results show that the effect of the intrinsic decoherence decreases the quasi-period of the entanglement between the atom and the field. The general conclusions reached are illustrated by numerical results.

- M Abdel-Aty, M Sebawe Abdalla and A-S F Obada, Entropy and phase properties of isotropic coupled oscillators interacting with a single atom: one-and two-photon processes,
*J. Opt. B: Quantum Semiclass Opt.***4**(2002), S133-S141.We investigate a modified model of the Jaynes-Cummings model involving multimode and multiphoton processes. The interaction part of the Hamiltonian consists of a two-level atom interacting with two isotropic fields injected simultaneously within a perfect cavity. The equation governing the system are solved and the relevant wave functions for one- and two-photon processes are obtained. In the two-photon case we invoked the SU (1, 1) Lie algebra to reach this solution. The density matrix elements are calculated and the entropy and the atomic inversion are computed. The influence of the mean photon numbers and the detuning on these quantities are studied. Comparison between the present model and the usual nonlinear model is given. The phase properties are also discussed.

- M Abdel-Aty, M Sebawe Abdalla and A-S F Obada, Entropy squeezing of a two-mode multiphoton Jaynes-Cummings model in the presence of a nonlinear medium,
*J. Opt. B: Quantum Semiclass Opt.***4**(2002), 134-142.From a quantum information point of view we study the entropy squeezing of a two-level atom interacting with two modes with intensity-dependent coupling. The Hamiltonian we consider consists of all acceptable forms of nonlinearities. A definition of squeezing is presented for this system, based on information theory. The utility of the definition is illustrated by examining squeezing in the information entropy of a two-level atom in the presence of a nonlinear medium. We examine the influence of the nonlinear interaction, the atomic coherence and the detuning parameter on the properties of the entropy and squeezing of the atomic variables. It is shown that features of the quantum entropy are influenced significantly by the kinds of intensity-dependent atom-field coupling and the nonlinearities of the two-mode fields.

- M Abdel-Aty and A-S F Obada, Induced emission of cold atoms passing through a micromaser cavity: spatial dependence,
*Mod. Phys. Lett. B***16**(2002), 117-125.The emission probability of a cold atom in a microcavity when its centre-of-mass motion is described quantum mechanically is presented, but is distinguished from other treatments by the inclusion of the spatial variation along the cavity axis. In particular, the mesa mode cavity profile is considered. The quantum theory of the one-photon mazer is constructed in the framework of the dressed-state coordinate formalism. Simple expressions for the atomic populations, the cavity photon statistics, and the reflection and transmission probabilities are given for any initial state of the atom-field system. The general conclusions reached are illustrated by numerical results.

- M Sebawe Abdalla, M Abdel-Aty and A-S F Obada, Quantum entropy of isotropic coupled oscillators interacting with a single atom,
*Optics Communications***211**(2002), 225-234.In this communication we introduce a new model of Hamiltonian which represents the interaction between a two-level atom and two electromagnetic fields injected simultaneously within a cavity. By using the canonical transformation the model can be regarded as a generalization of the Jaynes-Cummings model (JCM). The relationship between the original and new states has been established, and the exact expression for the wave function in Schrödinger picture is obtained. The atomic inversion and the degree of entanglement are considered. We find that the revival time is different from what has been observed in the JCM case.

- A-S F Obada and M Abdel-Aty, Entanglement degree of two two-level atoms,
*Information***5**(2) (2002), 191-202. - M Abdel-Aty, M Sebawe Abdalla and A-S F Obada, Uncertainty relation and information entropy of a time dependent bimodal two-level system,
*J. Phys. B: At. Mol. Opt. Phys.***35**(2002), 4773-4786.In this paper we consider a nonlinear interaction between a two-level atom and two modes of the electromagnetic field within a cavity. As the instantaneous position of the particle inside the cavity depends on both velocity and acceleration, we have assumed the coupling parameter to be explicitly time dependent. We have obtained the dynamical operators by solving the Heisenberg equations of motion and then constructed the density operator. The Heisenberg uncertainty relation is used to examine the entropy squeezing for this system. We have studied the effects of the phase, velocity and acceleration on entropy squeezing and atomic inversion. In addition the possibility of Bell- state production is discussed.

- A-S F Obada and H A Hessian, Influence of superposition of coherent states of light on the evolution of the field entropy and entanglement in the intensity-dependent JCM,
*Modern Physics Letters B***16**(28-29) (2002) 1097-1106.We investigate the influence of superposition of coherent states of light on population inversion and the evolution of the field entropy for two-level atoms in the framework of the Jaynes-Cummings model. We compare the behaviour of the system in the case of having a coherent superposition state and a statistical mixture of coherent states as an initial field. Our results show that the superposition of coherent states plays an important role in the evolution of the field entropy in the intensity-dependent Jaynes-Cummings model (JCM).

- T M El-Shahat, M Abdel-Aty and A-S F Obada, Do Useful Quantum Computing Models Exist?, P
*roc. Second Inter. Conf. on Information, Beijing, China 24-27 July 2002*(2002), 293-297. - M Abdel-Aty and A-S F Obada, Engineering entanglement of a general three-level system interacting with a correlated two-mode nonlinear coherent state,
*The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics***23**(1) (2003), 155-165.In this article a treatment of a three-level atom interacting with two modes of light in a cavity with arbitrary forms of nonlinearities of both the fields and the intensity-dependent atom-field coupling is presented. A factorization of the initial density operator is assumed, with the privileged field modes being in a pair-coherent state. We derive and illustrate an exact expression for the time evolution of the density operator, by means of which we identify and numerically demonstrate the region of parameters where significantly large entanglement can be obtained. We show that entanglement can be significantly influenced by different kinds of nonlinearities. The nonlinear medium yields the superstructure of atomic Rabi oscillation. We propose a generation of Bell-type states having a simple initial state preparation of the present system.

- A-S F Obada, A Schreiber and M Abdel-Aty, Entropic uncertainty relation of a single-trapped ion interacting with laser beams,
*Acta Physica Slovaca***53**(1) (2003), 37-47.In this communication we analyse the quantum motion of an ion isotropically confined in the radial plane of a Paul microtrap when it is irradiated by a properly chosen configuration of external laser beams. We present a derivation of an analytical solution, by means of which we study the entropic uncertainty relation. Using parameters corresponding to existing experimental realization, we have studied the variation effect for the initial total number of vibrational quanta and the phase.

- A-S F Obada, M Abdel-Aty, S Nakamura and S Furuichi, Quasi-mutual entropy of a general bimodal cavity,
*Int. J. Pure Appl. Math.***4**(2) (2003), 143-163.We develop a general formalism to describe the dynamical behaviour of an ensemble of a two-level system interacting with a bimodal cavity field including arbitrary forms of nonlinearities of both the field and the intensity-dependent coupling. We present a derivation of the unitary operator within the frame of the dressed state approach, by means of which we identify and numerically demonstrate the region of parameters where significantly large entanglement can be obtained. The influences of the nonlinearity, Stark shifts and detuning on the degree of entanglement are examined. It is shown that features of the degree of entanglement are influenced significantly by different kinds of nonlinearities. We propose a method of generating Bell-type states from a simple initial state preparation of two different modes of electromagnetic field. This investigation may be useful for developing non-linear optical devices, and providing some guidelines for experimentalists in the identification of an unknown non-linear medium and in the utilization of the nonlinearities of the atom-field interaction.

- M Sebawe Abdalla and A-S F Obada, Non-classical properties for some intermediate states,
*Arab J. Math. Sci.***9**(1) (2003), 29-105.The authors study several classes of non-classical states of light and analyse their non-classical properties, especially antibunching and non-Poissonian behavior. In particular, they consider even and odd generalizations of the binomial states (interpolating between coherent and number (Fock) states), the geometric states (interpolating between the Fock states and the thermal states) and the negative binomial states (interpolating between coherent states and quasi-thermal states). They study the Glauber second order correlation function, squeezing, the quasi probability distribution functions, with emphasis on the Wigner and the Q functions, and the photon number and the phase properties, as well. They study the dependence of these quantities on the variation of the parameters involved, as they interpolate between the limiting cases. Finally, they also present schemes for the production of these states and the application to the Jaynes-Cummings model and to resonance fluorescence in quantum optics.

- M Sebawe Abdalla, M Abdel-Aty and A-S F Obada, Entropy and entanglement of time dependent two-mode Jaynes-Cummings model,
*Phys. A***326**(1-2) (2003), 203-219.In this communication we examine the field entropy and the degree of entanglement for a time dependent Jaynes-Cummings model. We allow the amplitude of the laser field irradiating a trapped ion to be modulated. A sinusoidal modulation depending on a quadratic function in the argument is considered. At exact resonances the analytic solution for the wave function is given, and the density matrix operator is obtained. We show that for a small value of the acceleration the entanglement between the field and the atom would delay, while for a large value of the acceleration the entanglement decreases its value. We also show that the effect of the velocity is different from that of the acceleration where the value of the entropy decreases as we increase the value of its parameter.

- F A A El-Orany and A-S F Obada, On the evolution of superposition of squeezed displaced number states with the multiphoton Jaynes-Cummings model,
*J. Opt. B: Quantum Semiclass. Opt.***5**(1) (2003), 60-72.In this paper we discuss the quantum properties for superposition of squeezed displaced number states against the multiphoton Jaynes-Cummings model (JCM). In particular, we investigate atomic inversion, photon number distribution, purity, quadrature squeezing, the Mandel Q parameter and the Wigner function. We show that quadrature squeezing for the three-photon absorption case can exhibit revivals and collapses typical of those occurring in atomic inversion in the one-photon absorption case. Also, we prove that for odd number absorption parameter there is a connection between the evolution of the atomic inversion and the evolution of the Wigner function at the origin in phase space. Furthermore, we show that the non-classical states whose Wigner function values at the origin are negative will always be non-classical when they evolve through the JCM with even absorption parameter. We demonstrate that various types of cat states can be generated via this system.

- H H Salah, M Darwish and A-S F Obada, Generalized excited negative binomial states of electromagnetic field and some of their non-classical properties,
*International Journal of Modern Physics B***17**(7) (2003), 1071-1086.New states of electromagnetic field, generalized excited negative binomial states are introduced here. These states interpolate between the superposition of two excited coherent states and number states. The non-classical properties for these states are discussed, such as, second order correlation function, squeezing phenomena [normal squeezing and amplitude squared squeezing], phase properties in Pegg-Barnett formalism and the quasi-probability distribution functions (Q-function and Wigner function).

- A-S F Obada and M Darwish, A class of nonlinear coherent states and some of their properties,
*J. Opt. B: Quantum Semiclass. Opt.***5**(3) (2003), 211-217.Nonclassical properties such as the autocorrelation function, quadrature and amplitude-squared squeezing, the phase properties in the Pegg-Barnett formalism, the Husimi-Kano Q function and the Wigner-Moyal (W) function of a wider class of nonlinear coherent states, are calculated and discussed in this paper.

- A-S F Obada and M Abdel-Aty, Spatial variation effects on quantum theory of the micromaser with ultra-cold Lambda-type three-level atom,
*International Journal of Modern Optics B***17**(14) (2003), 2735-2747. - M Sebawe Abdalla, A-S F Obada and E M Khalil, Quantum Mechanics of Two-Field Parametric Frequency Converter Interacting with a Single Atom,
*International Journal of Theoretical Physics***42**(11) (2003), 2735-2763.In this communication we deal with a new model Hamiltonian representing the interaction between a two-level atom and two electromagnetic field modes in a cavity. The interaction between the modes has been taken into account and considered to be of a parametric frequency converter type. The model can be regarded as a generalization of two different systems: the Jaynes-Cummings model (atom-field interaction) and the two-mode frequency converter model (field-field interaction). Under a certain condition an exact solution for the equations of motion in the Heisenberg picture is given. The wave function in the Schrödinger picture is also constructed and used to discuss some statistical properties related to the model. We assume that the fields are initially in coherent states. We discuss atomic inversion, photon number distribution, squeezing and other phenomena. We show in all cases that the system is very sensitive to any variation in the mean photon numbers.

- A-S F Obada, M Sebawe Abdalla and E M Khalil, Statistical properties of two-mode parametric amplifier interacting with a single atom,
*Physica A: Statistical Mechanics and its Applications***336**(3-4) (2004), 433-453.In the present work we introduce a Hamiltonian model consisting of two fields injected simultaneously within a perfect cavity to interact with a single atom. The interaction between the fields has been taken into account and considered to be in the parametric amplifier form. The model can be regarded as a generalization of the Jaynes-Cummings model as well as a generalization of the parametric amplifier model. Under a certain condition, which is carefully selected, the exact solution for the Heisenberg equations of motion is obtained. Employing this solution we managed to discuss some statistical properties such as the atomic inversion, the photon number distribution, the squeezing phenomenon, the Glauber second-order correlation function and Phase distribution. Also the Q-function is considered.

- E M Khalil, M Sebawe Abdalla and A-S F Obada, Quantum treatment of two coupled oscillators in interaction with a two-level atom: parametric amplifier model,
*International Journal of Modern Physics B***18**(16) (2004), 2325-2345.In this communication we handle a modified model representing the interaction between a two-level atom and two modes of the electromagnetic field in a cavity. The interaction between the modes is assumed to be of a parametric amplifier type. The model consists of two different systems, one represents the Jaynes-Cummings model (atom-field interaction) and the other represents the two mode parametric amplifier model (field-field interaction). After some canonical transformations the constants of the motion have been obtained and used to derive the time evolution operator. The wave function in the Schrödinger picture is constructed and employed to discuss some statistical properties related to the system. Further discussion related to the statistical properties of some physical quantities is given where we have taken into account an initial correlated pair-coherent state for the modes. We concentrate in our examination on the system behaviour that occurred as a result of the variation of the parametric amplifier coupling parameter as well as the detuning parameter. It has been shown that the interaction of the parametric amplifier term increases the revival period and consequently longer period of strong interaction between the atom and the fields.

- M Sebawe Abdalla, A-S F Obada and M Abdel-Aty, Isotropic-coupled oscillators interacting with a single atom via two-photon processes: quantum information aspects,
*Journal of Physics B: Atomic, Molecular and Optical Physics***37**(4) (2004), 775-790.In the present paper we consider a nonlinear interaction between two electromagnetic field modes and a two-level atom. A new Hamiltonian model is introduced to represent such an interaction, where three coupling parameters are involved. We assume that the two-level atom interacts with each mode separately as well as with the two modes nonlinearly. Thus degenerate and non-degenerate processes are considered. Using a suitable canonical transformation and integrability condition the system is reduced to a single mode case. The density matrix is calculated and used to study the photon distribution, the atomic occupation probabilities, the von Neumann entropy and the entropy squeezing.

- A-S F Obada, S Furuichi, H F Abdel-Hameed and M Abdel-Aty, Pseudo information entropy of a single trapped ion interacting with a laser field,
*Inform. Sci.***162**(1) (2004), 53-61.In this communication, we use the pseudo entropy as a measure of information content of ionic state due to ion-laser interaction in a single trapped ion. We examine numerically the population inversion, and the entropy for the system. Minima for the degree of entanglement occur during the transition from maxima to minima (and vice-versa) for the atomic inversion. Successive peaks show a lowering of the local maximum point indicating a dissipative irreversible change in the ionic state. The results show that, the decoherence effect yields loss of information. The general conclusions reached are illustrated by numerical results.

- A-S F Obada and S Abdel-Khalek, New features of the atomic Wehrl entropy and its density in multi-quanta two-level system,
*J. Phys. A***37**(25) (2004), 6573-6585.This contribution describes new features of the atomic Wehrl entropy in a multi-quanta two-level system in the presence of the Kerr medium. A definition of the atomic Wehrl entropy is presented for this system, based on the atomic Q-function as information measurement about the atomic phase space. The influence of the nonlinear interaction of the Kerr medium, Stark shift, one and two-photon processes and the detuning parameters on the properties of the atomic Wehrl entropy is examined. The atomic Wehrl entropy gives equivalent results to the von Neumann entropy.

- A-S F Obada and H A Hessian, Entanglement generation and entropy growth due to intrinsic decoherence in the Jaynes-Cummings model,
*J. Opt. Soc. Amer. B Opt. Phys.***21**(8) (2004), 1535-1542.We study how intrinsic decoherence leads to growing entropy and a strong degradation of the maximal generated entanglement in the multiquanta Jaynes-Cummings model. We find an exact solution of the Milburn equation in multiquanta processes and calculate the partial entropy of the particle (atom or trapped ion) and field subsystem as well as the total entropy. As the total entropy is not conserved, and is shown to increase as time develops, one cannot use the partial field or atomic entropy as a direct measure of particle-field entanglement. For a good entropy measure, we also calculate the negativity of the eigenvalues of the partially transposed density matrix. We find that, at least qualitatively, the difference between the total entropy and the sum of field and atom partial entropies can also be used as an entanglement measure. Our results show that the degree of entanglement is very sensitive to any change in the intrinsic decoherence parameter.

- M Abdel-Aty and A-S F Obada, Analytic solution for entangled two-qubit in a cavity field,
*J. Math. Phys.***45**(11) (2004), 4271-4281.An exact solution of the time-dependent master equation that describes the evolution of two two-level qubits (ions or atoms) within a perfect cavity for the case of multiphoton transition and in the presence of both the Stark shift and phase shift is obtained. Employing this solution, the significant features of the entanglement when a second qubit is allowed to interact with cavity mode and becomes entangled with the first qubit are investigated in the context of the measure defined by negative eigenvalues for the partial transposition of the density operator. The effects of Stark shift, distance between the two qubits, and an instantaneous phase shift experienced by the second qubit on the entanglement and probability amplitudes are indicated. It has been shown that the entanglement as well as the intensity are markedly affected by different parameters when the nonlinear two-photon process is involved. Moreover, the quasi-probability distribution function is investigated before and after the sudden phase shift experienced by the second qubit. We believe that this may throw some light on the question of the entanglement of multi-qubit systems.

- A-S F Obada, M Darwish and H H Salah, Some non-classical properties of a class of new nonlinear coherent states,
*J. Modern Opt.***52**(9) (2005), 1263-1274.Some of the properties of a nonlinear coherent state associated with the centre of mass motion of trapped atoms are considered. In particular, the autocorrelation function, motional quanta number distribution, the phase properties, the Husimi-Kano Q-function and the Wigner-Moyal W-function of this nonlinear coherent state are discussed. Jumps in the motional quanta distribution function are reported in their dependence on the Lamb-Dicke parameters.

- M Sebawe Abdalla,; A-S F Obada and M Abdel-Aty, Von Neumann entropy and phase distribution of two mode parametric amplifier interacting with a single atom,
*Ann. Physics***318**(2) (2005) 266-285.In the present article, we introduce a Hamiltonian model that consists of two modes of the field in a perfect cavity to interact with a single two-level atom. The interaction between the fields has been taken into account and considered to be in the parametric amplifier form. The model in one hand can be regarded as a generalization of the Jaynes-Cummings model (JCM), however, in the other hand it can be considered as a generalization of the parametric amplifier model. Under a certain condition the exact solution to the Schrödinger equation is obtained. Employing this solution and for chosen values of different parameters we discuss numerically the atomic occupation probabilities as well as the degree of entanglement through the entropy of the field. The system shows superstructure phenomenon similar to that appeared from the effect of the Kerr-like medium on the Jaynes-Cummings model. The von Neumann entropy and phase distribution for both two-mode correlated and uncorrelated coherent states cases are also considered.

- A-S F Obada, H H Salah, M A Darwish and E M Khalil, Generalized trio coherent states,
*Internat. J. Theoret. Phys.***44**(9) (2005), 1347-1364.Generalized trio coherent states are made by superposing a number of trio coherent states. They possess inherent non-classical properties such as a sub-Poissonian distribution and violation of Cauchy-Schwarz inequalities. Their phase distribution is discussed in the framework of the Pegg and Barnett theory. We study the interaction of the radiation field prepared in a generalized trio coherent state with an atom in a superposition coherent state in the rotating wave approximation. We investigate the time dependence of the sub-Poissonian distribution and the violation of Cauchy-Schwarz inequalities for such a system. General conclusions reached are illustrated by numerical results.

- M Sebawe Abdalla, M Abdel-Aty and A-S F Obada, von Neumann mutual information for anisotropic coupled oscillators interacting with a single two-level atom,
*Internat. J. Theoret. Phys.***44**(9) (2005), 1649-1662.We consider the interaction between a two-level atom and two electromagnetic fields injected simultaneously within a cavity, with the interaction between the fields in parametric frequency-converter form. The wave function in the Schrödinger picture is obtained under certain conditions and consequently the density matrix. By employing a generalization of the von Neumann mutual information (in the context of Tsallis' non-extensive statistics) we measure the degree of entanglement for the present system. An important change is observed in the generalized mutual information depending on the entropic index. We also measure the minimum degree of entanglement during the transition from collapse to revival and vice-versa. Successive revival peaks show a lowering of the local maximum point indicating a dissipative irreversible change in the atomic state.

- A-S F Obada and M Darwish, Non-linear squeezing of the vacuum and the one-photon states as realizations of the SU(1,1) group,
*Journal of Optics B: Quantum and Semiclassical Optics***7**(2) (2005), 57-65.We introduce a class of non-linear squeezed vacuum and one-photon states. Under conditions on the non-linearity function, these states are realizations of the SU(1,1) group. However, when a condition is removed, these states are defined as eigenstates of certain operators. Some of the properties of these states are investigated for the case of trapped ions. In particular, the Mandel q number, the photon number distribution, the phase properties, the Husimi-Kano Q-function and the Wigner-Moyal W-function of the non-linear squeezed one-photon states are discussed.

- A-S F Obada and G M Abd Al-Kader, A class of nonlinear squeezed coherent states,
*Journal of Optics B: Quantum and Semiclassical Optics***7**(12) (2005), S635-S642.Nonlinear extensions of the single-mode squeezed vacuum and squeezed coherent states are constructed. Nonlinear squeezed coherent states (NLSCSs) are defined and special cases of these states are discussed. Two cases of the definition are considered for the unitary and non-unitary deformation operator functions. Some non-classical properties of these states are discussed. The Glauber second-order coherence function is calculated. Analytical and numerical results for the quadrature component distributions for the NLSCSs are presented. The s-parameterized quasi-probability function, especially the Wigner function, is discussed.

- M Sebawe Abdalla, A-S F Obada and M Darwish, Statistical properties of nonlinear intermediate states: binomial state,
*Journal of Optics B: Quantum and Semiclassical Optics***7**(12) (2005), S695-S704.In the present paper we introduce a nonlinear binomial state (the state which interpolates between the nonlinear coherent and number states). The main investigation concentrates on the statistical properties for such a state where we consider the squeezing phenomenon by examining the variation in the quadrature variances for both normal and amplitude-squared squeezing. Examinations for the quasi-probability distribution functions (W-Wigner and Q-functions) are also given for both diagonal and off diagonal terms. The quadrature distribution and the phase distribution as well as the phase variances are discussed. Moreover, we give in detail a generation scheme for such state.

- E M Khalil, M Sebawe Abdalla and A-S F Obada, Entropy and variance squeezing of two coupled modes interacting with a two-level atom: Frequency converter type,
*Annals of Physics***321**(2) (2006), 421-434,A modified Jaynes-Cummings model which consists of a two-level atom interacting with two modes of the electromagnetic field is introduced. More precisely we have considered a Hamiltonian model that includes two types of interaction: One is the field-field (frequency converter type) and the other is the atom-field interaction. By invoking a canonical transformation an exact solution of the wave function in the Schrödinger picture is obtained. The result presented in this context is used to discuss the atomic inversion as well as the entropy squeezing and variance squeezing phenomena. We have shown that the existence of the second field coupling parameter reduces the amount of squeezing in all quadratures, while the effect of the detuning parameter would lead to the superstructure phenomenon which becomes more pronounced upon increasing the mean photon numbers, in the states which are taken to be converter states.

- N Metwally and A-S F Obada, More efficient purifying scheme via controlled-controlled NOT gate,
*Physics Letters A***352**(1-2) (2006), 45-48.A new modified version of the Oxford purification protocol is proposed. This version is based on the controlled-controlled NOT gate instead of controlled NOT in the original one. Comparisons between the results of the new version and the original and an earlier modification are given. It is found that the new version converges faster and consumes fewer initial qubit pairs of low fidelity per final qubit pair of high fidelity.

- A-S F Obada, M Abdel-Aty, E M Khalil and G M Abd Al-Kader, Generation and some statistical properties of nonlinear pair-coherent states,
*Phys. Scr.***75**(4) (2007), 557-564.In this contribution a nonlinear type of the correlated two-mode states (nonlinear pair-coherent states (NLPCS)) is introduced. Based on the resonant ion-cavity interaction, we propose a scheme to generate these states revealing their connection with the pair-coherent state (PCS) and investigate their non-classical properties. The practical feasibility of this method is also discussed. The s-parameterized characteristic function (CF) of the NLPCS is considered. The s-parameterized quasi-probability distribution function (QDF) is investigated.

- M Sebawe Abdalla, M Abdel-Aty and A-S F Obada, Sensitive response of the quantum entropies to Jaynes-Cummings model in presence of a second harmonic generation,
*Internat. J. Theoret. Phys.***46**(3) (2007), 637-651.The behaviour of a modified Jaynes-Cummings Hamiltonian model (two-level atom in interaction with an electromagnetic field) in the presence of degenerate parametric amplification is introduced. We have examined two different cases, one when the field frequency wð is not equal to the splitting photon frequency µ for which the off-resonance case is considered. In the second case we have taken each frequencies to be equal (wð = µ) and considered the system to be at exact resonance. The wave function for both cases is obtained and the result used to calculate the density matrix from which we manage to discuss the field entropy as well as the phase entropy. It is shown that the system is sensitive to any change in the splitting photon frequency µ.

- G M Abd Al-Kader and A-S F Obada, Squeezed displaced Fock states in quantum optics,
*Int. J. Theor. Phys. Group Theory Nonlinear Opt.***11**(4) (2007), 207-263.The general classes of some quantum states are considered. Much attention is given to the squeezed displaced Fock states (SDFS's). These states can be regarded as a generalization of the number, coherent, squeezed coherent, squeezed number and displaced Fock (number) states. Some definitions for the SDFS's are studied. The superposition of these quantum states are reviewed. A detailed discussion of a type of four-component superposition of SDFS's is presented. The non-classical properties of these states such as photon number distribution and squeezing are discussed. Non-diagonal P-representation with both displaced Fock and squeezed states basis are reviewed. The s-parameterized characteristic function (CF) for the SDFS's is given. The s-ordered distribution functions of the SDFS's are investigated. The s-parameterized CF for the superposition of SDFS's is given. The s-parameterized quasi-probability distribution function for the superposition of SDFS's are investigated. Various moments are calculated by using this characteristic function. The Glauber second-order coherence function is calculated. The photon number distribution of the superposition of SDFS's are studied. Analytical and numerical results for the quadrature component distributions for the superposition of a pair of SDFS's are presented. The Pegg-Barnett phase distribution is given and compared with the radial integration of the s-parameterized distribution function over the phase space variable of SDFS's. Some applications for using the SDFS's as an input state of the interaction with atoms are discussed: The input field of the linear amplifier is assumed to be in a SDFS's. The Fokker-Planck equation for the damped harmonic oscillator is considered, and its steady state solution is given. A generation scheme using the trapped ions center of mass approach is discussed. The displaced two-photon realization of SU(1,1) Lie algebra is employed to define the analytic representation based on SDFS's. The interaction of these states with a two-level atom in a cavity with the presence of additional Kerr medium is studied. Exact general matrix elements of the time-dependent operators of a Jaynes-Cummings model, in the presence of a Kerr medium, with these states are derived. The atomic inversion and photon number distribution are discussed.

- M Sebawe Abdalla, E M Khalil and A-S F Obada, Statistical properties of a two-photon cavity mode in the presence of degenerate parametric amplifier,
*Annals of Physics***322**(11) (2007), 2554-2568.A Hamiltonian model that includes two-photon interaction with a two-level atom and a degenerate parametric amplifier is considered. By invoking a canonical transformation an exact solution of the wave function in the Schrödinger picture is obtained. The result presented in this context is employed to discuss the purity, the entropy squeezing, and the variance squeezing, in addition to the normal squeezing. It has been shown that the existence of the second harmonic generation leads to reduction in the squeezing amount for all quadrature variances and we found that as the value of the coupling parameter lð_2 increases the squeezing phenomenon gets more apparent. Further we have also considered the Q-function as an example of the quasi-probability distribution.

- A-S F Obada, H A Hessian and A-B A Mohamed, Entropies and entanglement for decoherence without energy relaxation in a two-level atom,
*Journal of Physics B: Atomic, Molecular and Optical Physics***40**(12) (2007), 2241-2248.In this paper, we investigate the temporal evolution of various measures of degree of entanglement and total correlation of a single-mode light field interacting with a two-level atom. The phase-damped cavity is taken into account. The effects of phase damping on a temporal of partial entropies for the atom and the field, total entropy, the entropy difference and the sum of negative eigenvalues of the partially transposed density matrix as a quantitative entanglement measure for the master equation in the dressed-state approximation are taken into account.

- A-S F Obada and M Abdel-Aty, Quantitative aspects of entanglement in optically driven quantum dots,
*Physics Review B***75**(2007), 195310-1-195310-7.We present an approach to look for the existence of maximum entanglement in a system of two identical quantum dots coupled by the Förster process and interacting with a classical laser field. Our approach is not only able to explain the existing treatments but also provides further detailed insights into the coupled dynamics of quantum-dot systems. The result demonstrates that there are two ways of generating maximum entangled states, one associated with far off-resonance interaction and the other associated with the weak-field limit. Moreover, it is shown that exciton decoherence results in the decay of entanglement.

- M Sebawe Abdalla, A-S F Obada and S Abdel-Khalek, Entropy squeezing of time dependent single-mode Jaynes-Cummings model in presence of non-linear effect,
*Chaos Solitons & Fractals***36**(2) (2008), 405-417.A general Hamiltonian for a system which consists of a two-level atom interacting with electromagnetic field is considered. Besides the Stark shift the model includes multi-photon process as well as a non-linear Kerr-like medium effect. Furthermore the field and the atom are assumed to be coupled with modulated coupling parameter which depends explicitly on time. Under an approximation where fast oscillations are ignored, an exact solution for the wave function in Schrödinger picture is given. The atomic inversion as well as the entropy squeezing are considered and it has been shown that existence of the time dependent coupling parameter leads to a time delaying in the interaction which is twice the delay time for the independent case.

- A-S F Obada, A A Eied and G M Abd Al-Kader, Entanglement of a general formalism -type three-level atom interacting with a non-correlated two-mode cavity field in the presence of nonlinearities,
*Journal of Physics B: Atomic, Molecular and Optical Physics***41**(19) (2008), 195503-1-195503-9.We investigate the evolution of the atomic quantum entropy and the atom-field entanglement in a system of a -configuration three-level atom interacting with a two-mode field with additional forms of nonlinearities of both the field and the intensity-dependent atom-field coupling. With the derivation of the unitary operator within the frame of the dressed state and the exact results for the state of the system we perform a careful investigation of the temporal evolution of the entropy. A factorization of the initial density operator is assumed, considering the field to be initially in a non-correlated two-mode squeezed coherent or binomial state. The effects of the mean photon number, detuning, Kerr-like medium and the intensity-dependent coupling functional on the entropy are analysed.

- A-S F Obada, H A Hessian and A-B A Mohamed, Entropy and entanglement in the Jaynes-Cummings model with effects of cavity damping,
*Journal of Physics B: Atomic, Molecular and Optical Physics***41**(13) (2008), 135503-1-135503-7.The temporal evolution of entanglement for a single-mode field interacting with a two-level atom via intensity-dependent coupling in the off-resonant case has been studied, where the leakage of photon through the cavity is taken into account. The effects of cavity damping on the coherence properties of the atom and the field are studied. The amount of entanglement is compared with the total correlations. It is found that the atom-field system is inhibited from going into a pure state in the off-resonant case.

- A-S F Obada, M M A Ahmed, E M Khalil and S I Ali, Finite Dimensional Trio-Coherent States and Some Non-Classical Properties,
*Chinese Journal of Physics***46**(4) (2008), 479-491.A new state of the electromagnetic field is introduced. It is a correlated three-mode state but in finite dimensions. Based on the resonant ion-cavity interaction, we propose a scheme to generate these states, revealing their connection with the nonlinear three-mode interaction. A relation to the realization of the SU(2) algebra is sought. We also investigate some of their non-classical properties.

- A-S F Obada, A A Eied and G M Abd Al-Kader, Treatment of the emission and absorption spectra of a general formalism -type three-level atom driven by a two-mode field with nonlinearities,
*Journal of Physics B: Atomic, Molecular and Optical Physics***41**(11) (2008), 115501-1-115501-12.An analytical expression of the emission and absorption spectra, for a -type three-level cavity-bound atom interacting with a two-mode cavity field, is given using the dressed states of the system. We take explicitly into account the existence of forms of nonlinearities of both the field and the intensity-dependent atom-field coupling. The characteristics of the emission and absorption spectra for binomial and squeezed coherent states of the modes are exhibited. The effects of the mean number of photons, detuning and the nonlinearity forms on the spectra are analysed.

- G M Abd Al-Kader and A-S F Obada, On SU(1,1) intelligent coherent states,
*Physica Scripta***78**(3) (2008), 035401-1-035401-10.Generalized coherent states associated with SU(1,1) Lie algebra are reviewed. A state is called intelligent if it satisfies the strict equality in the Heisenberg uncertainty relation. The eigenvalue problem satisfied by intelligent states (IS) is solved. The IS associated with SU(1,1) Lie algebra are investigated. We have constructed some realizations for our results of IS, and some applications are discussed. Some non-classical properties such as Glauber second-order correlation function, photon number distribution and squeezing are investigated.

- M Sebawe Abdalla, A-S F Obada and M Darwish, Non-classical properties of a nonlinear generalized geometric state,
*Physica Scripta***77**(5) (2008), 055002-1-055002-11.In this paper, we introduce and investigate some properties of a nonlinear generalized geometric state (the state that interpolates between the number state and a nonlinear pure thermal state). We mainly concentrate on the statistical properties. We have discussed the normal squeezing as well as the amplitude squared squeezing, further the Mandel's q-parameter is also considered. The investigation is also extended to include the quasi-probability distribution functions (W-Wigner and Q-functions). The quadrature distribution and the phase properties in the Pegg-Barnett formalism besides the phase variances are considered.

JOC/EFR May 2019

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