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Trapped and laser-cooled ions are increasingly used for a variety of modern high-precision experiments, frequency standard applications and quantum information processing. Therefore, in this communication we present a comprehensive analysis of the pattern of information entropy arising in the time evolution of an ion interacting with a laser field. A general analytic approach is proposed for a three-level trapped-ion system in the presence of the time-dependent couplings. By working out an exact analytic solution, we conclusively analyse the general properties of the von Neumann entropy and quantum information entropy. It is shown that the information entropy is affected strongly by the time-dependent coupling and exhibits long time periodic oscillations. This feature attributed to the fact that in the time-dependent region Rabi oscillation is time dependent. Using parameters corresponding to a specific three-level ionic system, a single beryllium ion in a RF-(Paul) trap, we obtain illustrative examples of some novel aspects of this system in the dynamical evolution. Our results establish an explicit relation between the exact information entropy and the entanglement between the multi-level ion and the laser field. We show that different nonclassical effects arise in the dynamics of the ionic population inversion, depending on the initial states of the vibrational motion/field and on the values of Lamb-Dicke parameter η.

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... Hence, our model can be (is) a qutrit-quadrit entangled system. For the high degree of entanglement, we report the quantum dynamics using quantum entropy (Abdel-Aty 2005) in the presence of the coupling parameters. ...

... This work focuses on the amount of quantum entanglement in system with the quantum entropy. A measurement of quantum entanglement for pure qudit states can be obtained using a generalized entropy Dermez et al. (2013);von Neumann (1995); Abdel-Aty (2005). ...

In this work, we investigate the quantum entangled qudits in a trapped three-level ion combining two photons in $\Lambda$ configuration setting a 12-dimensional Hilbert space. First, the quantum entropy (E) was examined and established as a ”developed measure”. Quantum correlations and entanglement were characterized with E. Second, different rates of the set of Lamb-Dicke Parameter (LDP) values were explored. Third, the entropy values for four optimum scaled times were given as a table. The findings showed when that $\eta$ increased, the sudden death of entanglement in the trapped ion-phonon system decreased, vice versa. For entangled qudits, such as qubit-qutrit and qutrit-quadrit, the sudden birth of entanglement can be tuned by LDP. We have demonstrated that quantum entanglement is stored in our system. Moreover, the quantity of quantum entropy is maximum degree of $E=0.706$.

... Several studies have been done on atom-atom and atomsfield entanglement. In such systems, generalizations like multiphoton transition, intensity-dependent coupling, multi-mode field, Kerr-medium, Stark shift, etc. have been studied in recent decades [19][20][21][22][23][24] . For example, Abdel-Aty [20] , have considered the effect of Stark shift played on entanglement in a double two-photon Jaynes-Cummings model (JCM). ...

... In such systems, generalizations like multiphoton transition, intensity-dependent coupling, multi-mode field, Kerr-medium, Stark shift, etc. have been studied in recent decades [19][20][21][22][23][24] . For example, Abdel-Aty [20] , have considered the effect of Stark shift played on entanglement in a double two-photon Jaynes-Cummings model (JCM). They have been found that the so-called Entanglement Sudden Death (ESD) can happen under suitable conditions related to the Stark effect for a specific initial state of the quantum system. ...

In this paper, two-level atoms interacting with a single-mode radiation field alongside important properties are presented. The model presented discusses multi-photon process and it has also a nonlinear Kerrmedium and Stark-shift. In addition, the coupling parameter is discussed in a time-dependent fashion.
The results show that Stark-shift, time-dependent coupling parameter, and nonlinear Kerr-medium play
significant roles regarding the evolution of the von-Neumann entropy and the system’s geometric phases.
We’ve used accessible parameters to examine these observations, and some new results are obtained.
Remarkably, the geometric phase shows high sensitivity to any change of the considered parameters.
© 2019 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved

... 6. The upper bound of S H (t) in this manuscript 6.1 and by comparing this result with [112,113], we may conclude that the upper bound of Shannon entropy is related to the initial number of photonsn by the relation S M ax ≤ ln(n) = =1 lnn s.t.n > 1 and is the number of field modes. So, we may reformulate the definition of Shannon information entropy as S H (t) = 1 =1 lnn ln P(n 1 , n 2 , t) P(n 1 ,n 2 ,t) ...

... • The upper bound of S H (t) in this manuscript 6.1 and by comparing this result with [112,113], we may conclude that the upper bound of Shannon entropy is related to the initial number of photonsn by the relation S M ax ≤ ln(n) = =1 lnn s.t.n > 1 and is the number of field modes. So, we may reformulate the definition of Shannon information entropy as S H (t) = 1 =1 lnn ln P(n 1 , n 2 , t) P(n 1 ,n 2 ,t) ...

... The option of rapid modification of the parameters of the AFS has experimental application in quantum information in previously never reach regimes, thus allowing the investigation of a large change of new physical systems (Bennett and Di Vincenzo 2000;Khalil 2013). Furthermore, information entropy as an entanglement indicator of a time-dependent three-level trapped ion interacting with a laser field has been discussed (Abdel-Aty 2005, 2003aAbdel-Khalek et al. 2010;Abdel-Aty 2007;El-Shahat et al. 2003b). More general treatments of the interaction between a two-level atom and cavity field in arbitrary forms of nonlinearities (El-Shahat et al. 2003b;Abdel-Aty 2003a, b;Abdel-Khalek et al. 2010;Abdel-Aty 2007, 2002Sebawe Abdalla et al. 2005;Abdel-Aty et al. 2002b, c). ...

The interaction of a 2-level atom with a time-dependent cavity field (two-photon non-degenerate transitions) in the parametric amplifier-type is considered. Using the integrability conditions, taking into account the phase and coupling, a general solution of the model is obtained through the Schrödinger equation. The effects of the functional dependence of the coupling and the initial state of the 2-level atom on atomic inversion, the degree of entanglement, entropy squeezing are investigated. It is shown that the acceleration term plays an important role in controlling the functions behavior of the considered quantities.

... Finally, all of these calculations in this paper can be treated with another quantum systemslike in three and four-level systems as in Refs. [58][59][60]. ...

We investigate some new properties of the model of a pure bipartite system in dissipative environments. The mathematical model of bipartite system with the interaction cavity field in the presence of dissipative environment is designed. The systems is assumed to be initially in the Warner state. The system is analytically solved using master equation method. We quantified the dynamics of quantum correlation between the system, under the influence of the mean photon numbers, type of the initial state and spontaneous emission rate of atom using several measures are, measurement-induced disturbance (MID) and Geometric measure of quantum discord (GQD), negativity (N) and concurrence (C). The effect of the system parameters on dynamics of entanglement is investigated, it is found that the spontaneous emission rate of atom number have much effect on all measures. Moreover the Geometric measure of quantum discord (GQD) have different behavior rather than the others and more influenced by the mean photon number and type of the initial state.

... 13 Additionally, many aspects of entanglement dynamics have been discussed. [14][15][16][17][18][19][20][21][22][23][24] On the other hand the entanglement properties of qubit systems under the influence of the environments have been investigated for different types of spin-paths. For example, the decoherence of a single qubit and entanglement of two qubits interacting with the spin-path modeled by XXZ spin chain, 25 the suppression of decoherence in a generalization of a spin-path model, two spatially separated qubits coupled to a common heat path and entangled by purely dissipative dynamics, 26 entanglement generation via a completely mixed nuclear spin-path [27][28][29] and entanglement sharing and decoherence in the spin-path. ...

We analyze a crucial effect of the spin-path environment on a single and maximum entangled two-qubit systems. For a single qubit, we investigate the coherent loss by means of coherent-vectors' dynamics and the interacted qubits' fidelity. We used entanglement and population dynamics to investigate the coherent loss of the two-qubit system. We show and numerically verify that the effect of the detuning and coupling parameters in the negativity can be mapped onto the maximum and minimum values of the entanglement.

... It is well known that entanglement represents the corner stone of most applications of quantum information [1,2,3]. There are several studies devoted to investigate the possibility of generating entanglement between different types of particles [4,5,6]. Quantifying the degree of entanglement which may be generated between these particles assimilates another area for many researchers, where several measurements of entanglement have been found. ...

Effect of Lorentz transformation on some properties of multi-qubit systems is
investigated. It is shown that, properties like, the fidelity and entanglement
decay as the Wigner's angles increase, but can be improved, if all the
transformed particles are transformed with the same Wigner's angles. The upper
bounds of the average capacity of the GHZ state increases while it decreases
and more robust with the W-state as the Wigner's angle of the observer
decreases. Under Lorentz transformation, the tripartite states transform into
another equivalent states and hence no change on the efficiency of these states
to perform quantum information tasks.

... Trapped ions can be well isolated from the environment and easily manipulated with lasers [17]. The simplest entangled states of qudits have been successfully used for decades as the main tool for testing the essentials of quantum mechanics [13,18,19]. If long-living entanglement is created it can be used for higher-capacity quantum information resources. ...

In this work, the quantum entanglement and quantum linearity in a trapped ion–phonon system are theoretically explained in terms of the second-order terms of the Lamb–Dicke parameter, η. For the trapped ion–phonon system, the optimum times are 2 and 10 and the optimum probability amplitudes of ions are 0.4 and 0.58, respectively. With the optimum times and amplitudes, it is shown that as η grows, the quality of the entanglement in the trapped ion–phonon system increases. In particular, we have demonstrated the quantum linearity due to second-order terms of η.

... The above results allows us to identify the dependence of these features on different parameters of the system, and it would interesting to consider a multi-level system [24,25], two-mode [26] or nonlinear interaction [27]. Furthermore, the quantum features of many systems decay uniformly as the result of decoherence and much effort has been directed to extend the coherence time of these qubits. ...

In this paper, we investigate the dynamics of coded information in a single Cooper pair interacting with a single Cavity mode. The effect of the relative ratio of Josephson junction capacity and the gate capacities on the purity, coherent vectors and the entropy of the traveling Cooper pair are investigated. The exchange information between the environment and the Cooper pair is quantified for different values of the Cooper qubit and environment parameters.

In this letter, we have proposed a new model for quantum control of atom photon entanglement in a single layer graphene via von Neumann reduced entropy of entanglement. We consider the effect of terahertz laser field intensity on the degree of entanglement (DEM) in the resonance and off-resonance condition of the applied fields. We also investigate the spatially dependent of the DEM when two applied light becomes standing wave pattern in x and y directions. Our results show that in different parametric conditions, the population of the different states can be controlled and this leads to modifying the DEM of the system.

This research examines the effect of an open system containing the squeezed generalized amplitude damping channel on the joint remote preparation quantum communication protocol using a maximally entangled two-qubit state. Our findings indicate that the fidelity of a quantum system in contact with a non-zero temperature thermal bath can be enhanced by varying the squeezing parameters. These parameters include the squeezing phase of the channel Φ$\Phi$ and the amount of squeezing of the channel r.

The goal of this paper is to identify the consequences of Darcy–Forchheimer flow (DFF) on electromagnetohydrodynamic flow of graphene oxide–iron oxide hybrid nanofluid over a rotating disk in a porous medium with viscous dissipation. The set of obtained ordinary differential equations had been solved with the corresponding boundary conditions using a numerical method called fourth-order Runge–Kutta method along with the shooting technique. The impact of the pertinent parameters on the dimensionless flow and temperature field profiles is shown using graphs. Also the nondimensional skin friction factor is stated in tabular form. The results state that as there is an increase in the value of porosity parameter, the velocity profile then diminishes. As shown in the outcomes, we accomplish that in this modeling, platelets have higher influence than the blade, brick, and cylinder. Due to nanoparticles, graphene oxide–iron oxide nanocomposite exhibits anti-microbial capabilities. These studies suggest that graphene oxide–iron oxide nanocomposite may be used to remove natural solvents and water filter.

Background and objective
Non-Newtonian (Reiner-Rivlin) nanofluid is novel prospective in various biology and biomedical (medicine) engineering processes and in many other biological science. In addition, various biological liquids, such as saliva, synovial liquid, blood have non-Newtonian behavior and can demonstrate important viscoelastic characteristics. This treatise elucidates broad feature of cavitation in non-Newtonian liquids and bubble dynamics and applies them to the fields of bioengineering and biomedicine. In view of such biomedical and bioengineering applications the aim of this paper is to discuss the hydromagnetic flow of Reiner-Rivlin nanomaterial subject to a rotating disk is addressed. Disk rotates with constant angular frequency about vertical axis. Reiner’s equation of a general viscous fluid differs from the Navier’s Stokes equation by a quadratic expression describing cross-viscosity coefficient. Energy equation is obtained using thermal radiation and heat generation. Thermodynamics second law investigates the entropy. Multiple slip conditions are analyzed. Thermophoresis and random diffusion behaviors are addressed. First order reaction is also taken into account.
Methodology
Nonlinear systems are reduced to dimensionless system by employing similarity transformation. ND-solve procedure is implemented on Mathematics software for the computation of nonlinear differential system.
Results
Graphical representation of velocity, temperature and concentration are obtained. Entropy generation has been emphasized. Velocity has decaying trends for magnetic variable. Random diffusion variable have opposite trends for concentration and temperature. An intensification in thermal slip variable reduces temperature.

We study the dynamics of global quantum discord and von Neumann entropy for systems composed of two, three, and four two-level atoms interacting with the single-mode coherent field under the influence of a nonlinear Kerr medium. The collapses and revivals of the global quantum discord and von Neumann entropy are observed for different values of the nonlinear Kerr medium parameter for both initial pure and mixed states. Moreover, it is found that at higher values of the nonlinear Kerr medium parameter, the magnitude of the revivals of quantum entanglement is suppressed. It is also noted that for mixed states global quantum discord shows comparatively damped oscillations as compared to the pure states. It is worth mentioning that by increasing the average number of photons in the system, quantum entanglement and quantum discord exhibit damping behavior. Furthermore, the revival time of both the global quantum discord and von Neumann entropy increase with the increase in the nonlinear Kerr medium parameter, for the systems with a relatively large number of atoms.

The theme of this article is to address the irreversibility in an incompressible Reiner-Rivlin nanofluid subject to stretchable rotating disk. Dissipation and radiation in heat expression are incorporated. Random diffusion and thermophoresis impacts are addressed. Physical feature of entropy rate is also accounted. Furthermore, first order reaction rate is scrutinized. Ordinary system (ODEs) is obtained through implementation of suitable variables. To construct convergent solution, we employed numerical method (ND-solve method). Outcomes for flow variables on velocity profile, thermal field, concentration and entropy optimization are discussed. Computational outcomes of moment coefficient, skin friction coefficient, entrainment velocity (disk pumping efficiency), Sherwood number and gradient of temperature versus sundry variables are studied. An expansion in radial velocity is observed for Reiner-Rivlin fluid variable. An augmentation in stretching parameter leads to opposite behavior of radial and tangential velocity components. An amplification in temperature distribution and entropy rate are observed for radiation variable. An improvement in thermophoresis parameter augments concentration and temperature distribution. Higher approximation of Brinkman number rises entropy generation rate.

A quantum scheme is presented by which a three-level trapped ion interacts with a two laser beams in the absence and presence of the e®ect of classical field. We analyze the impact of the classical ¯eld and the Lamb-Dicke parameter (LDP) on the dynamical behavior of entanglement quanti¯er, population probabilities and the geometric phase. Based on four different variations of these two effects, LDP = 0.1, LDP=0.01 and ¯ = 0.0, ¯ = 0.49, the time dependence of geometric phase and populations probabilities are shown. The ¯nding emphasizes that both the time-dependent and LDP play an important role in the development of the entanglement, the geometric phase, ¯delity, and populations probabilities. This in-sight may be very useful in various applications in quantum optics and information processing.

Background and objective
Here irreversibility analysis in unsteady Darcy-Forchheimer flow of viscous fluid by a stretched sheet is examined. Lorentz force effect is considered. Energy attribution is discussed through thermodynamic first law with Joule heating, radiation and dissipation effects. Entropy generation is calculated. Thermo diffusion and diffusion-thermo characteristics are also examined. Furthermore binary chemical reaction is addressed. Significance of entropy generation and heat transfer rate is considered. Here our main aim is to discuss the entropy rate and heat transfer analysis. The recommended model is pertinent for heat exchangers, entropy optimization, biomedicine, two-phase flows, polymers, thermal and solutal transportation, fuel cells and geothermal energy system.
Methods
Partial differential equations (PDEs) are altered into dimensionless form through appropriate variables. The resulting governing equations are then solved numerically by using finite difference method (FDM).
Results
Influence of sundry variables on velocity, temperature, entropy optimization and concentration are deliberated. Skin friction coefficient and Sherwood and Nusselt numbers are scrutinized. An improvement in velocity field is noticed for Reynold number. An increment in magnetic parameter has reverse trend for temperature and velocity. Temperature is augmented against higher radiation and Dufour number. Concentration has opposite characteristics with variation of reaction parameter and Soret number.
Conclusions
Significance enhancement in velocity gradient is seen for higher magnetic and porosity variables. An increment in Forchheimer number improves the surface drag force. Magnitude of Nusselt number (heat transfer rate) reduces for higher Reynold number. An improvement in radiation declines the heat transfer rate. An augmentation in radiation parameter improves the temperature and entropy generation rate. Entropy generation rate is augmented versus Reynold number and radiation parameter. Entropy optimization and surface drag force have similar effects for magnetic parameter.

A clear relationship between the quantum version of Fisher information and the bipartite system entanglement has been explored recently. Known existing results for some quantum systems having strongly nonlocal correlations are still limited. In this paper, based on the interaction between a two-qubit and squeezed field, we investigate the correlation between the dynamical behavior of qubit-qubit entanglement and the quantum Fisher information (QFI). We focus on the effect of a number of photon multiplicity, low and high squeezing regime of the field on the dynamics of the QFI and entanglement quantifier. We show how the QFI and entanglement quantifier can be affected by the number of photon multiplicity and squeeze parameter during the time evolution of cases of absence and presence of dissipation effect. Our results exhibit the correlation between the entanglement quantifiers and the QFI with respect to the appearance of collapse and revival phenomena of the atomic inversion.

In this paper, we study a model of two two-level atoms interacting with a quantum field. An analytical solution is obtained which is used to study the information entropy of the system. It is shown that the nonlinear term plays a significant role in the behaviour of the minimum uncertainty (MU) compared with the concurrence (C). Our extensive study of information entropy of atoms–field interaction demonstrates that using the coupling strength between the atoms and the field as a controller parameter, one can control the dynamics of the system by increasing the lower bound of the entropic uncertainty relation or decreasing the entanglement.

In the language of the complex formalism, we study the information entropy of a particle on the motion groups from a family of the unitary Cayley-Klein space with constant curvature κ. Hence, in making use of the constant curvature, all the results here presented will be simultaneously valid for the 2D coset space SUκ(2)/U(1) in forms of 2D sphere ${{\mathbb{S}}}^{2}(\kappa \gt 0)$ , hyperbolic plane ${{\mathbb{H}}}^{2}(\kappa \lt 0)$ and Euclidean plane ${{\mathbb{E}}}^{2}(\kappa =0)$ . In addition to physical complex coordinates $({z}_{0},{\bar{z}}_{1})$ , their corresponding components $({p}_{{z}_{0}},{\bar{p}}_{{z}_{1}})$ are expressed in a 2D complex formalism in terms of the constant curvature in Cayley-Klein space. This process enables us to derive information entropies located in the circular well on two spaces, which are the basis for achieving the relationship between Shannon entropy and Fisher information and the inequalities among them. In particular, we notice that the particle has freely behavior in a spherical state ${{\mathbb{S}}}^{2}$ , while it behaves as a constraint particle in the situation of the hyperbolic plane ${{\mathbb{H}}}^{2}$ .

In this paper, we present some properties through two two-level atoms interacting with a two-mode quantized cavity field. We study this system in the presence of detuning parameter, Kerr nonlinearity, Stark shift, relative phase and intensity-dependent atoms-field coupling. Also, the coupling parameter is modulated to be time dependent. The exact solution of this model is given by using the Schrődinger equation when the atoms and the field are initially prepared in superposition states and coherent states, respectively. We employed the results to calculate some aspects such as linear entropy, total atomic inversion and cross-correlation function.

The entangled qudits of three-level trapped ion and two phonons (in coherent state) in $\Lambda$ configuration forming a Hilbert space of 12-D are investigated. The quantum entropy is analyzed “such as an elaborated measure” in trapped ion-coherent state system. Four values of Lamb-Dicke parameter (LDP), $\eta =0.005, 0.07, 0.08$ and 0.09 are probed for deep Lamb-Dick regime. We elucidate that as $\eta$ is increased,
sudden death of entangled state in the trapped ion-coherent state system is decreased or vice versa. By this way, sudden birth of entangled state can be tuned by LDP. All graphs in this study are plotted with aid of the Wolfram Mathematica 9.

In previous studies, information dynamics methods such as Von Neumann entropy and Rényi entropy played an important role in many fields, covering both macroscopic and microscopic studies. They have a solid theoretical foundation, but there are few reports in the field of mechanical nonlinear systems. So, can we apply Von Neumann entropy and Rényi entropy to study and analyze the dynamic behavior of macroscopic nonlinear systems? In view of the current lack of suitable methods to characterize the dynamics behavior of mechanical systems from the perspective of nonlinear system correlation, we propose a new method to describe the nonlinear features and coupling relationship of mechanical systems. This manuscript verifies the above hypothesis by using a typical chaotic system and a real macroscopic physical nonlinear system through theory and practical methods. The nonlinear vibration correlation in multi-body mechanical systems is very complex. We propose a full-vector multi-scale Rényi entropy for exploring the chaos and correlation between the dynamic behaviors of mechanical nonlinear systems. The research results prove the effectiveness of the proposed method in modal identification, system dynamics evolution and fault diagnosis of nonlinear systems. It is of great significance to extend these studies to the field of mechanical nonlinear system dynamics.

In this manuscript, we present a system consisting of a three-level atom interacting with optical field. We investigate qualitatively the entanglement and atomic (field) geometric phase under the effect of cavity damping. The atom–field entanglement is measured by the negativity. We show that these quantifiers depend strongly on the variations of the initial settings of the atom, and this exhibits substantial phenomena that depend on the cavity damping effect. Finally, we explore the link between the entanglement and atomic (field) geometric phase of different physical parameters within the presence and absence of the cavity damping effect.

In this paper, we consider an explicit solution of system of Sylvester matrix equations of the form A1V1 − E1V1F1 = B1W1 and A2V2 − E2V2F2 = B2W2 with F1 and F2 being arbitrary matrices, where V1,W1,V2 and W2 are the matrices to be determined. First, the definitions, of the matrix polynomial of block matrix, Sylvester sum, and Kronecker product of block matrices are defined. Some definitions, lemmas, and theorems that are needed to propose our method are stated and proved. Numerical test problems are solved to illustrate the suggested technique.

In this paper, we consider a three-level Λ-type atom interacting with a two-mode of electromagnetic cavity field surrounded by a nonlinear Kerr-like medium, the atom and the field are suffering decay rates (i.e. the cavity is not ideal) when the multi-photon processes is considered. Also, the atom and the field are assumed to be coupled with a modulated time-dependent coupling parameter under the rotating wave approximation. The wave function and the probability amplitudes are obtained, when the atom initially prepared in the superposition states and the field initially in the coherent states, by solving the time-dependent Schrödinger equation by taking a proper approximation to the system of differential equations. An analytical expression of the atomic reduced density operator is given. We studied the degree of entanglement, between the field and atom, measure (DEM) via the concurrence, Shannon information entropy, momentum increment and diffusion, and finally we investigated the effects of decay rates and the time-dependent parameters on Husimi Q-function.

In this paper, the dynamical properties of a trapped three-level ion which interacts with a two-mode quantized field in the presence of a Kerr medium are studied. The ion-field coupling is also considered to be intensity-dependent (nonlinear interaction). In this respect, the
Λ
, V and ladder-type three-level ions are considered. Via solving the time-dependent Schörodinger equation, the state vectors of the corresponding ion-field systems are analytically obtained after choosing special initial conditions. In the continuation, the mean phonon number as a criterion of trapping energy, the population inversion as a criterion of exchanging energy (between the ion and the field) and the linear entropy as a criterion of entanglement between the trapped ion and the field are studied. To get more general insight about the interaction conditions, the effect of intensity-dependent ion-field coupling (nonlinearity regime) on the dynamics of system is investigated. Finally, by comparing the obtained results for the three configurations of trapped three-level ions with the previously considered two-level one, we conclude that the degree of entanglement for three-level ions is more than the two-level ion. Also, among the considered trapped ions, the ladder-type ion possesses the most amount of entanglement with the quantized field, whereas the
Λ
-type possesses the least. We also conclude that, the general temporal behaviour of the other considered quantities (mean phonon number and population inversion) are the same for all three types of trapped three-level ions.

In this paper, we study the dynamics of the atomic inversion and von Neumann entropy for a moving and non-moving two-level atom interacting with multi SU(1,1) quantum system. The wave function and system density matrix using specific initial conditions are obtained. The effects of initial atomic state position and detuning parameters are examined in the absence and presence of the atomic motion effect. Important phenomena such as entanglement sudden death, sudden birth and long-living entanglement are explored during time evolution. The results show that the detuning parameter and excitation number is very useful in generating a high amount of entanglement.

The effect of atomic spontaneous decay is considered for a two-level atom interacting with a single mode of electromagnetic field.The exact solution of the master equation is found for a chosen initial state. We study the effects of atomic decay on information and entanglement through temporal evolution of atomic quantum Fisher information, partial entropy of the atom and negativity.

In this paper, the Wehrl entropy approach is discussed and compared with the quantum entanglement using a mixed-state three-level atom interacting with a cavity field. In the pure state case, the behavior of the atomic Wehrl entropy shows the same behavior of the entanglement due to the von-Neumann entropy, while the mixed state case gives the total correlation due to quantum mutual entropy. If the system is in an entangled state, the formalism can be used to quantify the entanglement as well as the total correlations.

We solved time-dependent Hamiltonian of a three-level ion interacting with two-laser beams in Λ scheme using a unitary transformation method which transform the Λ scheme to a cascade Ξ scheme for the vibrational phonon transitions. Our analytical results for the probability amplitudes become precise after a certain period of time when the Lamb-Dicke regime is reached. We analyzed the entanglement created in the system.

We give a theoretical description of two time-dependent laser beams in the Λ scheme using a unitary transformation method and a trapped three-level ion. We extend earlier investigations aimed at finding the three types of density matrices. We present figures showing that the entanglement degree accelerates due to the time-dependent interaction and the second-order terms of the Lamb–Dicke parameter η(t). Our results explain that the time-dependent ionic–phononic quantum system is observed at a higher degree of entanglement for three optimum times; these are, respectively, 16.5, 110, and 220 fs. These optimum entangled states can be modified for the structure of black holes in a probabilistic Universe.

Our aim is to investigate the entanglement dynamics and quantum correlations of a full-trapped ion interacting with two time-independent laser beams in view of the Lamb–Dicke parameter. For this purpose, the three probability amplitudes in the trapped ion is taken as
$\sqrt {{{{1} \left/ {3} \right.}}}$
. Concurrence, negativity, and atomic Wehrl entropy of entanglement exhibit a long interacting time. We show that long survival is proved with these quantum measures.

We study the interaction between a moving two-level atom and a single-mode field. The coupled atom–cavity system with atomic center-of-mass motion included is modeled by considering the dependence of the atomic motion along z-axis. At exact resonance between the internal atomic transition and the cavity eigenfrequency, an exact solution of the system is obtained and periodically modulated Rabi oscillations and regular translational motion are observed. We focused on the dynamics of both field Wehrl entropy and Wehrl phase distribution. The influence of the atomic motion on the evolution of von Neumann entropy and Wehrl entropy is examined. The results show that the atomic motion and the field-mode structure play important roles in the evolution of the von Neumann entropy, Wehrl entropy and Wehrl PD.

We investigate a scheme for an environment-induced geometric phase based on a multi-level artificial atom, initially prepared in a mixed state, that is sent through a superconducting cavity. It is shown that the environment provides a mechanism for the generation of a long-lived geometric phase. Owing to the effect of the decoherence, the solution of the master equation allows one to calculate the W-function for the multi-level system. In this regime the negativity disappears due to a stronger decoherence.

We construct a complete representation of the atomic information entropy of an arbitrary multi-level system. Our approach is applicable to all scenarios in which the quantum state shared by a single particle and fields is known. As illustrations we apply our findings to a single four-level atom strongly coupled to a cavity field and driven by a coherent laser field. In this framework, we discuss connections with entanglement frustration and entropic forms. We conclude by showing how the atomic information entropy can be extended to examine entanglement in multi-level atomic systems.

We propose the use of atomic Wehrl entropy associated to the reduced atomic density operator as an entanglement indicator of bipartite systems. This is applied to a two-level system (one single harmonically trapped ion) by taking into account the linear center-of-mass-motional degree of freedom. Detailed analytical and explicit expressions are given, taking into account different configurations. The results show the important roles played by the laser phase and initial state setting in the evolution of the atomic Q-function, atomic Wehrl entropy and marginal atomic Q-function. Our procedure of using atomic Wehrl entropy may be applied to a system with Hilbert space of high dimension.

We investigate the dynamics of a tri-beam laser interacting with a closed three-level atomic system. The Bloch equations are obtained using Hesinberg’s equation of motion. The impact of the resulting parameters on the atomic occupation probabilities and information entropy is discussed. It is shown that the populations are affected by the detuning in different ways which are then compared in order to simulate realizations entanglement through the information entropy.

We analyze different entanglement measures for a mixed state two-level system in the presence of intrinsic decoherence. The information about entanglement is obtained by comparing the results for the atomic Wehrl entropy and negativity with the analytical results for a simple case. For the strong decoherence case we find that a similar and long-lived maximum Wehrl entropy and negativity between atom and field are shown. The results highlight the important roles played by both the decoherence parameter and the initial state setting in determining the evolution of the atomic Wehrl entropy and negativity.

The dynamics of the skew information (SI) is investigated for a single Cooper
Pair Box,CPB interacts with a single cavity mode. The effect of the cavity and
CPB's parameters on the SI is discussed. We show that, it is possible to
increase the skew information to reach its maximum value either by increasing
the number of photons inside the cavity or considering non-resonant case with
larger detuning parameter. The effect of the relative ratio of Josephson
junction capacity and the gate capacity is investigated, where the number of
oscillations of the skew information increases by decreasing this ratio and
consequently the travelling time between the maximum and minimum values
decreases.

In this paper, the problem of a three-level atom interacting with a single mode of the quantized electromagnetic cavity field is considered. The considered theoretical model describes the distinct configurations of a three-level atom. Also, this model includes a detuning and arbitrary forms of both the field and the intensity-dependent atom-field coupling. We obtain the general forms for both the constants of motion and the wavefunction when the atom is initially prepared in one of its states. We use this model for computing a number of statistical aspects of the three-level system. As an illustration, we use the model for studying the time evolution of the second-order correlation function when the atom is initially in an upper state and the field is considered in a coherent state.

Higher dimensional quantum entanglement in a trapped three-level ion interacting with two laser beams in Λ scheme is investigated beyond the Lamb–Dicke limit. It is shown that higher dimensional entanglement can be established in a single step, with a tunable dimensionality and duration via the Lamb–Dicke parameter.

We study the interaction of a V-type three-level atom with a two-mode field through the two-photon interaction when the atom is initially in the upper state
and the initial field is in the two-mode squeezed vacuum state. The influence of the atomic motion on the evolution of the
atomic Q-function and atomic Wehrl entropy is examined. The results show that the atomic motion and the mode structure play important
roles in the evolution of the atomic Q-function, atomic Wehrl entropy, and marginal atomic Wehrl entropy.

We analyze the atomic Wehrl entropy and negativity as compared with concurrence for qudit pure states in a trapped ion. We
use the density matrix in calculating the three measures of quantum correlations. We find that a long surviving entangled
qudit can be established between the three atomic levels and vibrational modes. We observe three distinct entanglements in
response to an increasing Lamb–Dicke parameter.
Keywordsconcurrence–quantum entanglement–trapped ions

In this paper we are interested in studying the entanglement between a single four-level ladder-type atom interacting with
one-mode cavity field when the atomic motion is taken into account. The exact solution of the model is obtained by using Schrodinger
equation for a specific initial conditions. The field entropy of this system is investigated in the non-resonant case. The
effects of the detuning parameter and the atomic motion on the entanglement degree are examined. These investigations show
that both of the detuning and the atomic motion play important roles in the evolution of the von Neumann entropy and atomic
populations. Finally, conclusions and some features are given.
KeywordsFour-level atom–von Neumann entropy–Entanglement

Motivated by recent developments in quantum entanglement, we study the relations among concurrence and phase entropy of a
three-level atom interacting with a bimodal cavity field. Analytical results are presented when the photonic band gap is exhibited
by the presence of photonic crystals. The evolution of the atomic inversion with the field initially in a coherent state is
examined, and different nonclassical effects in its dynamics are discussed. An extension of the notion of concurrence introduced
by Wooters is used to quantify the entanglement. We conclusively calculate the phase entropy and entanglement using the Pegg-Barnett
phase formalism. Evidence has been found to support the idea that phase entropy and concurrence are correlated in this particular
model. One feature of the regime considered here is that closed-form evaluation of the time evolution may be carried out in
the presence of the detuning and the photonic band gap, which provides insight into the difference in the nature of the concurrence
function for atom-field coupling, mode frequency, and different cavity parameters. We demonstrate how fluctuations in the
concurrence and phase entropy are affected by the presence of the photonic band gap. Explicit results with numerical simulations
applied to GaAs are obtained.

Quantum entangled states in a system of trapped three-level ion
interacting with two laser beams in Λ (Lambda)
configuration is investigated. We have characterized a typical
family of initial conditions for their potential to generate
quantum entanglement of internal and external degrees of freedom
of the ion. It is found that entangled qudits, specifially qutrits
and quadrits, can be optimally for a certain preparation of the
ionic system. Analytical results, describing the quantum entangled
state explicity, are presented. The amount of quantum entanglement
is quantified directly by calculating the generalized concurrence
for arbitrary qudits. It is obtained that higher dimensional
entanglement can be established with the Lamb-Dicke parameter
(LDP). The LDP dependence of Schmidt coefficients is shown.

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.

The uncertainty principle in quantum physics for non-commuting observables can be quantitatively expressed in various mathematical forms. Well known are common Heisenberg-like uncertainty relations in which the product of dispersions of two non-commuting observables is used. An alternative form of uncertainty relations represents the so-called entropic uncertainty relations in which the sum of entropies of the considered non-commuting observables is employed. The aim of this paper is to show that the entropic uncertainty relations exhibit several interesting mathematical properties and can be taken as an adequate alternative formulation of the uncertainty principle in quantum mechanics. To compare the mathematical properties of the common and entropic uncertainty relations we present some simple quantum systems with two non-commuting observables.
Dedicated to Dr B Mamojka on the occasion of his fiftieth birthday.
Abstrakt. Princip neurcitosti pro nekomutující veliciny v kvantové fyzice mze být matematicky vyjádren r formou. Dobre známé jsou relace neurcitosti Heisenbergova typu, v nichz se vyuzí vá soucinu disperzí dvou nekomutujících fyzikálních velicin. Alternativou jsou tzv. entropické relace neurcitosti, v nichz vystupuje soucet entropií uvazovaných velicin. Cílem tohoto clánku je ukázat, ze tyto relace mají nekolik z matematického hlediska zajímavých vlastností a mohou být pouzity k alternativnímu vyjádrení principu neurcitosti v kvantové mechanice. Srovnání vlastností Heisengergových a entropických relací neurcitosti je demonstrováno na jednoduchém kvantovém systému s dvema nekomutujícími velicinami.

We introduce a family of separability criteria that are based on the existence of extensions of a bipartite quantum state rho to a larger number of parties satisfying certain symmetry properties. It can be easily shown that all separable states have the required extensions, so the nonexistence of such an extension for a particular state implies that the state is entangled. One of the main advantages of this approach is that searching for the extension can be cast as a convex optimization problem known as a semidefinite program. Whenever an extension does not exist, the dual optimization constructs an explicit entanglement witness for the particular state. These separability tests can be ordered in a hierarchical structure whose first step corresponds to the well-known positive partial transpose (Peres-Horodecki) criterion, and each test in the hierarchy is at least as powerful as the preceding one. This hierarchy is complete, in the sense that any entangled state is guaranteed to fail a test at some finite point in the hierarchy, thus showing it is entangled. The entanglement witnesses corresponding to each step of the hierarchy have well-defined and very interesting algebraic properties that, in turn, allow for a characterization of the interior of the set of positive maps. Coupled with some recent results on the computational complexity of the separability problem, which has been shown to be NP hard, this hierarchy of tests gives a complete and also computationally and theoretically appealing characterization of mixed bipartite entangled states.

A new class of uncertainty relations is derived for pairs of observables in a finite-dimensional Hilbert space which do not have any common eigenvector. This class contains an ``entropic'' uncertainty relation which improves a previous result of Deutsch and confirms a recent conjecture by Kraus. Some comments are made on the extension of these relations to the case where the Hilbert space is infinite dimensional.

Among the numerous types of architecture being explored for quantum computers are systems utilizing ion traps, in which quantum bits (qubits) are formed from the electronic states of trapped ions and coupled through the Coulomb interaction. Although the elementary requirements for quantum computation have been demonstrated in this system, there exist theoretical and technical obstacles to scaling up the approach to large numbers of qubits. Therefore, recent efforts have been concentrated on using quantum communication to link a number of small ion-trap quantum systems. Developing the array-based approach, we show how to achieve massively parallel gate operation in a large-scale quantum computer, based on techniques already demonstrated for manipulating small quantum registers. The use of decoherence-free subspaces significantly reduces decoherence during ion transport, and removes the requirement of clock synchronization between the interaction regions.

Experiments directed towards the development of a quantum computer based on trapped atomic ions are described briefly. We discuss the implementation of single-qubit operations and gates between qubits. A geometric phase gate between two ion qubits is described. Limitations of the trapped-ion method such as those caused by Stark shifts and spontaneous emission are addressed. Finally, we describe a strategy to realize a large-scale device.

The phenomenon of atomic population trapping in the Jaynes-Cummings Model is analysed from a dressed-state point of view. A general condition for the occurrence of partial or total trapping from an arbitrary, pure initial atom-field state is obtained in the form of a bound to the variation of the atomic inversion. More generally, it is found that in the presence of initial atomic or atom-field coherence the population dynamics is governed not by the field's initial photon distribution, but by a `weighted dressedness' distribution characterising the joint atom-field state. In particular, individual revivals in the inversion can be analytically described to good approximation in terms of that distribution, even in the limit of large population trapping. This result is obtained through a generalisation of the Poisson Summation Formula method for analytical description of revivals developed by Fleischhauer and Schleich [Phys. Rev. A {\bf 47}, 4258 (1993)].

Methods for, and limitations to, the generation of entangled states of trapped atomic ions are examined. As much as possible, state manipulations are described in terms of quantum logic operations since the conditional dynamics implicit in quantum logic is central to the creation of entanglement. Keeping with current interest, some experimental issues in the proposal for trapped-ion quantum computation by I. Cirac and P. Zoller (University of Innsbruck) are discussed. Several possible decoherence mechanisms are examined and what may be the more important of these are identified. Some potential applications for entangled states of trapped-ions which lie outside the immediate realm of quantum computation are also discussed.

This communication is an enquiry into the circumstances under which concurrence and phase entropy methods can give an answer to the question of quantum entanglement in the composite state when the photonic band gap is exhibited by the presence of photonic crystals in a three-level system. An analytic approach is proposed for any three-level system in the presence of photonic band gap. Using this analytic solution, we conclusively calculate the concurrence and phase entropy, focusing particularly on the entanglement phenomena. Specifically, we use concurrence as a measure of entanglement for dipole emitters situated in the thin slab region between two semi-infinite one-dimensionally periodic photonic crystals, a situation reminiscent of planar cavity laser structures. One feature of the regime considered here is that closed-form evaluation of the time evolution may be carried out in the presence of the detuning and the photonic band gap, which provides insight into the difference in the nature of the concurrence function for atom-field coupling, mode frequency and different cavity parameters. We demonstrate how fluctuations in the phase and number entropies effected by the presence of the photonic-band-gap. The outcomes are illustrated with numerical simulations applied to GaAs. Finally, we relate the obtained results to instances of any three-level system for which the entanglement cost can be calculated. Potential experimental observations in solid-state systems are discussed and found to be promising. Comment: 28 pages, 10 figures: Accepted in Applied Physics B: Laser and Optics

An approach to the seperability problem of a bipartite system based on the entropic uncertainty relations was proposed. The proposed strategy took advantage of the geometrical structure of the tensor product Hilbert space of the system and underlines the connections between uncertainty relations and entanglement. To replace the statistical variance with the Shannoy entropy as an estimator of the uncertainities associated with the measurement process was the basic idea of the proposed approach. It was shown that entropic uncertainty relations can be derived for more than two observables at a time.

We have investigated motional heating of laser-cooled 9Be+ ions held in radio-frequency (Paul) traps. We have measured heating rates in a variety of traps with different geometries, electrode materials, and characteristic sizes. The results show that heating is due to electric-field noise from the trap electrodes which exerts a stochastic fluctuating force on the ion. The scaling of the heating rate with trap size is much stronger than that expected from a spatially uniform noise source on the electrodes (such as Johnson noise from external circuits), indicating that a microscopic uncorrelated noise source on the electrodes (such as fluctuating patch-potential fields) is a more likely candidate for the source of heating. Comment: With minor changes. 24 pages, including 7 figures. Submitted by Phys. Rev. A

The author provides, in nine chapters, a compact and up-to-date coverage of the entire range of applications of the two most important ion traps: the Paul('radiofrequency') trap and the Penning('dc) trap. The book begins with full details of the ion confinement principles of both these traps; this is followed by a presentation of the basic experimental techniques, including details of a few actual traps. There is then a chapter on the methods of ion cooling, now an essential integral part of all trap-based experiments. The next four chapters provide a comprehensive coverage of applications in four major areas, broadly classified as: atomic physics, frequency standards, collision studies, and analytical mass spectrometry. The text is appended by a set of more than 600 fully titled chronologically arranged references which mirror the growth of the field as well as providing a comprehensive guide to original research papers. The text should be useful to students both at the senior undergraduate and beginning graduate level as a general reader for professionals in atomic physics, chemical physics, mass spectometry and related fields.

The usual definition of squeezing, based on the Heisenberg uncertainty principle, measures uncertainty in terms of the standard deviation. It can run into difficulties when applied to squeezing in the two-level atom. An alternative definition of squeezing is presented for this system, based on information entropy theory, which overcomes the disadvantages of the definition based on the Heisenberg uncertainty relation. The utility of this definition is illustrated by examining squeezing in the information entropy of a two-level atom in the Jaynes-Cummings model, and in resonance fluorescence.

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 atmoic 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 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.

We show that, if one combines the Jaynes-Cummings and anti-Jaynes-Cummings dynamics in a trapped-ion system driven by a laser, additional series of collapses and revivals of the vibrational state of the ion can be generated.

The object of this Letter is to show that except in the case of canonically conjugate observables, the generalized Heisenberg inequality does not properly express the quantum uncertainty principle. It is, in general, too weak. An inequality is obtained which does express the principle.

This Letter reports on the existence of periodic spontaneous collapse and revival of coherence in the dynamics of a simple quantum model. Also given are the first accurate expressions for the intermediate-time and long-time dynamical behavior of the model.

In this work, the time evolution of the entropy of a single-mode field interacting with a Λ-type degenerate quantum beat three-level atom is investigated in the strong field limit. The field is assumed to be prepared in a coherent state. The results show that the initial atomic state and the detuning of the field play important roles in the evolution of the field entropy and the generation of a Schrödinger cat state at the half-revival time of the atomic inversion needs strict conditions. The squeezing properties of the field are also explored, and some new conclusions are obtained.

A theorem is proven for quantum information theory that is analogous to the noiseless coding theorem of classical information theory. In the quantum result, the von Neumann entropy S of the density operator describing an ensemble of pure quantum signal states is equal to the number of spin-1/2 systems (‘‘quantum bits’’ or ‘‘qubits’’) necessary to represent the signal faithfully. The theorem holds whether or not the signal states are orthogonal. Related results are also presented about the fidelity of quantum coding and about representing entangled quantum states.

We investigate quantum dynamical properties of a trapped three-level ion interacting with two laser beams in Λ configuration. A unitary transformation method is developed to study the interaction of the ion with its vibrational phonons, quanta of ion’s own center-of-mass motion. Under certain conditions on laser parameters, this interaction is shown to be unitarily equivalent to two-phonon cascade transitions. Complicated temporal behaviors of level populations and mean number of phonons are described clearly by identifying dynamical variables of the cascade model as building blocks. Furthermore, analyzing quantum states of vibrational phonons by Husimi-Q function, we find that at times, determined by the underlying cascade dynamics, two- and three-component macroscopic quantum superposition states can be obtained depending on the Lamb-Dicke parameter and the initial conditions of the system. A wide range of initial conditions and experimental parameters are discussed using both exact and analytical solutions. Alternative routes to reach the target states are found.

In this paper, we use the quantum field entropy to measure the degree of entanglement in the time development of a three-level atom interacting with two-mode fields including all acceptable kinds of nonlinearities of the two-mode fields. This model describes a very general situation in which the system admits two detuning parameters and includes an arbitrary form of the nonlinearity of the two modes. It is shown that when the individual modes of the field are detuned far from the intermediate atomic level, the dynamic Stark shift is induced by the nonlinear medium. The results show that the non-linearity effect yields the superstructure of atomic Rabi oscillations and changes the quasiperiod of the field entropy evolution and entanglement between the atom and the field. The general conclusions reached are illustrated by numerical results.

We have investigated the evolution of the field quantum entropy and the entanglement of the atom field in a three-level atom, 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, where the privileged field mode is 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 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 quasiperiod of the field entropy evolution and entanglement between the atom and the field. The general conclusions reached are illustrated by numerical results.

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.

The number-phase entropic uncertainty relation for the
multiphoton coherent state (MCS) and nonlinear coherent
state (NCS) are studied and compared with an ordinary coherent
state (CS). We show that the MCS has higher (lower) number (phase)
entropy while the NCS has lower (higher) number (phase) entropy
in comparision to the CS. We also discuss the
number-phase Wigner function for these states which gives
a graphical representation of the complimentary nature of the
number and phase properties for these states. With the help of
this Wigner representation a photon number cat formation in the
NCS is discussed.

Analytic expressions have been found for transition probabilities in a degenerate
n-level
atom interacting with a strong external field that gives a common time dependence
to all of the transition matrix elements. Except for solving a simple nth-order
equation to determine eigenvalues of dressed states, the method is entirely analytic.
These expressions may be used to control electron populations in degenerate
n-level atoms. Examples
are given for n = 2
and 3.

In the present communication we investigate the usual Jaynes–Cummings Hamiltonian model, describing two-level atom interacting with an electromagnetic field, in the presence of the second harmonic generation (degenerate parametric amplifier). Exact solutions of the wave function in the Schrödinger picture have been obtained for two different cases. In the first case the field frequency ω is not equal to the splitting photon frequency ε, where the canonical transformation has been invoked to obtain the solution of the wave function. In the second case, we considered both frequencies are equal (ε = ω) and the system is taken to be at exact resonance. Both solutions have been used to discuss the atomic inversion as well as the entropy squeezing. It has been shown that the system is sensitive to any change in the coupling parameter responses of the second harmonic generation as well as to the atomic phase angle.

We present theoretical and experimental studies of the center-of-mass c.m. stability of ions in a Penning trap with a quadrupole rotating electric field. The rotation frequency of an ion cloud in a Penning trap determines the cloud density and shape, and it can be precisely controlled by a rotating electric field. The quadrupole rotating-field scheme can control pure single-species plasmas in contrast to the dipole field, which is effective only for plasmas composed of two or more species of ions. However, the quadrupole field can modify the trap stability because of the spatial dependence of the electric field. In this study, we theoretically and experimentally determine the c.m. stability condition for ions in a Penning trap with a rotating quadrupole field. The experimental results agree well with the theoretical prediction. In the limit of zero magnetic field we obtain a type of rf trap which uses a rotating quadrupole field and in which the c.m. motion is analytically solvable.

The exact upper and lower bounds on the sum of the information entropies of three spin-1/2 operators SX, SY, SZ are derived. They show that, for a set of more than two observables, entropic uncertainty and certainty relations can exist which do not reduce to those satisfied by the pairs in the set. This result is generalized to sets of N+1 complementary observables existing in N-dimensional Hilbert spaces. Publicado

The exact lower bound on the sum of the information entropies is obtained for arbitrary pairs of observables in two-dimensional Hilbert space. The result coincides with that given by Garrett and Gull for the particular case of real transformation matrices and state vectors. A weaker analytical bound is also obtained. Publicado

The entropic uncertainty relation for sets of $N+1$ complementary observables $\{A_k\}$ in $N$-dimensional Hilbert space, $\sum_kH(A_k)\geq (N+1)\ln[\frac12(N+1)]$, is sharpened to $\sum_kH(A_k)\geq\frac12N\, \ln(\frac12N)+(\frac12 N+1)\!\ln(\frac12N+1)$ for even $N$. A nontrivial upper bound on the entropy sum (entropic certainty relation) is also obtained for not completely mixed states, while a previously given expression for this bound is proved to hold only when $N=2$. Publicado

Considering a system consisting of a two-level atom, initially prepared in a coherent superposition of upper and lower levels, interacting with a coherent state of the field, we show that the dynamics of the atom as well as the spectrum of the field are sensitive to the relative phase between the atomic dipole and the cavity field. It is shown that, for a certain choice of this phase, ``coherent trapping'' occurs in two-level atoms. In the case of spectra, for the same choice of the phase, instead of a three-peaked symmetric spectrum, we have an asymmetric two-peaked spectrum.

We investigate the phenomenon of quantum revivals in the Jaynes-Cummings model for an arbitrary quantized field mode. With the help of the Poisson summation formula, we cast the infinite sum determining the atomic inversion into an infinite sum of integrals. Each integral, when evaluated using the method of stationary phase, yields under appropriate conditions one revival. We present simple approximate analytical expressions for these revivals and illustrate this general technique by the examples of a coherent and a highly squeezed state. The oscillatory photon distribution of the latter creates slightly different Rabi frequencies which give rise to a beat note; that is, echos in the revivals. We obtain the photon statistics of the quantized field by ``measuring'' the atomic collapse of a single revival-a technique which might be applicable in the realm of the one-atom maser.

We propose a technique to generate nonclassical vibrational states of the quantized center-of-mass motion of an ion in a harmonic trap, based on the quantum conversion between the quantum cavity field and the quantized center-of-mass motion. It is shown that when an ion trap system interacts with the eigenmode of a single-mode Fabry-Pérot cavity, where the trap is set in, and with an external classical driving electromagnetic field through Raman transitions, the interchange of the quantum features between the quantum cavity field and the quantized trap occurs. This kind of quantum conversion can be used to prepare some nonclassical trap states for the ion trap as well as to measure the quantum statistics of an initial nonclassical vibrational state.

A quantum computer can be implemented with cold ions confined in a linear trap and interacting with laser beams. Quantum gates involving any pair, triplet, or subset of ions can be realized by coupling the ions through the collective quantized motion. In this system decoherence is negligible, and the measurement (readout of the quantum register) can be carried out with a high efficiency.

We report the creation of thermal, Fock, coherent, and squeezed states of motion of a harmonically bound {sup 9}Be{sup +} ion. The last three states are coherently prepared from an ion which has been initially laser cooled to the zero point of motion. The ion is trapped in the regime where the coupling between its motional and internal states, due to applied (classical) radiation, can be described by a Jaynes-Cummings-type interaction. With this coupling, the evolution of the internal atomic state provides a signature of the number state distribution of the motion. {copyright} {ital 1996 The American Physical Society.}

We reconstruct the density matrices and Wigner functions for various quantum states of motion of a harmonically bound {sup 9}Be{sup +} ion. We apply coherent displacements of different amplitudes and phases to the input state and measure the number state populations. Using novel reconstruction schemes we independently determine both the density matrix in the number state basis and the Wigner function. These reconstructions are sensitive indicators of decoherence in the system.

Quantum computers require the storage of quantum information in a set of two-level systems (called qubits), the processing of this information using quantum gates and a means of final readout. So far, only a few systems have been identified as potentially viable quantum computer models--accurate quantum control of the coherent evolution is required in order to realize gate operations, while at the same time decoherence must be avoided. Examples include quantum optical systems (such as those utilizing trapped ions or neutral atoms, cavity quantum electrodynamics and nuclear magnetic resonance) and solid state systems (using nuclear spins, quantum dots and Josephson junctions). The most advanced candidates are the quantum optical and nuclear magnetic resonance systems, and we expect that they will allow quantum computing with about ten qubits within the next few years. This is still far from the numbers required for useful applications: for example, the factorization of a 200-digit number requires about 3,500 qubits, rising to 100,000 if error correction is implemented. Scalability of proposed quantum computer architectures to many qubits is thus of central importance. Here we propose a model for an ion trap quantum computer that combines scalability (a feature usually associated with solid state proposals) with the advantages of quantum optical systems (in particular, quantum control and long decoherence times).

I propose a scheme which allows for reliable transfer of quantum information between two atoms via an optical fibre in the presence of decoherence. The scheme is based on performing an adiabatic passage through two cavities which remain in their respective vacuum states during the whole operation. The scheme may be useful for networking several ion-trap quantum computers, thereby increasing the number of quantum bits involved in a computation. Comment: 4 pages, 3 figures, RevTeX, submitted to PRL

We propose a scheme to utilize photons for ideal quantum transmission between atoms located at spatially-separated nodes of a quantum network. The transmission protocol employs special laser pulses which excite an atom inside an optical cavity at the sending node so that its state is mapped into a time-symmetric photon wavepacket that will enter a cavity at the receiving node and be absorbed by an atom there with unit probability. Implementation of our scheme would enable reliable transfer or sharing of entanglement among spatially distant atoms. Comment: 4 pages, 3 postscript figures

We discuss the relationship between entropic uncertainty relations and entanglement. We present two methods for deriving separability criteria in terms of entropic uncertainty relations. Especially we show how any entropic uncertainty relation on one part of the system results in a separability condition on the composite system. We investigate the resulting criteria using the Tsallis entropy for two and three qubits. Comment: 8 pages, 3 figures, v2: small changes

In this tutorial we review physical implementation of quantum computing using a system of cold trapped ions. We discuss systematically all the aspects for making the implementation possible. Firstly, we go through the loading and confining of atomic ions in the linear Paul trap, then we describe the collective vibrational motion of trapped ions. Further, we discuss interactions of the ions with a laser beam. We treat the interactions in the travelling-wave and standing-wave configuration for dipole and quadrupole transitions. We review different types of laser cooling techniques associated with trapped ions. We address Doppler cooling, sideband cooling in and beyond the Lamb-Dicke limit, sympathetic cooling and laser cooling using electromagnetically induced transparency. After that we discuss the problem of state detection using the electron shelving method. Then quantum gates are described. We introduce single-qubit rotations, two-qubit controlled-NOT and multi-qubit controlled-NOT gates. We also comment on more advanced multi-qubit logic gates. We describe how quantum logic networks may be used for the synthesis of arbitrary pure quantum states. Finally, we discuss the speed of quantum gates and we also give some numerical estimations for them. A discussion of dynamics on off-resonant transitions associated with a qualitative estimation of the weak coupling regime and of the Lamb-Dicke regime is included in Appendix.

We first consider the basic requirements for a quantum computer, arguing for
the attractiveness of nuclear spins as information-bearing entities, and light
for the coupling which allows quantum gates. We then survey the strengths of
and immediate prospects for quantum information processing in ion traps. We
discuss decoherence and gate rates in ion traps, comparing methods based on the
vibrational motion with a method based on exchange of photons in cavity QED. We
then sketch the main features of a quantum computer designed to allow an
algorithm needing 10^6 Toffoli gates on 100 logical qubits. We find that around
200 ion traps linked by optical fibres and high-finesse cavities could perform
such an algorithm in a week to a month, using components at or near current
levels of technology.