## Vol 19, No 2 (2015)

**Year:**2015**Articles:**13**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1218

Articles

Entanglement of two qubits interacting with one-mode quantum field

#### Abstract

In the present paper we investigate the dynamics of the system of two twolevel natural or artiﬁcial atoms, in which only one atom couples to a thermal one-mode ﬁeld in ﬁnite-Q cavity, since one of them can move around the cavity. For the description of the dynamics of the system we ﬁnd the eigenvalues and eigenfunctions of a Hamiltonian of the system. With their help we derive the exact expression for a density matrix of the system in case of a pure initial state of atoms and a thermal state of a ﬁeld. The reduced atomic density matrix is found. The one-qubit transposing of an atomic density matrix is carried out. With its help the Peres-Horodecki criterium is calculated. Numerical calculations of entanglement parameter is done for different initial pure states of atoms and mean photon numbers in a thermal mode. It is found that the thermal ﬁeld can induce a high degree of qubits entanglement in considered model. Thus we have derived that one can use the strength of dipole-dipole interaction and cavity temperature for entanglement control in the considered system. It is shown also that the maximum degree of entanglement is reached for one-atom excited state.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(2):205-220

The quantum transitions probability as paths-integral in energy states space

#### Abstract

By the use of the functional integration approach (paths integral approach) we present a non-perturbative method for dynamics of multi-levels quantum systems (such as atoms, molecules and nanosystems) interacting with high-intensity laser ﬁeld describing. The probability of transitions between investigated quantum system states under electromagnetic ﬁeld action is written as functional integral in energy representation (in investigated quantum system energy states space). In this approach we calculate probabilities of diatomic molecules transition between rotating quantum states under the ultrashort laser pulses train action by the use of numerical simulations. We investigate the dynamics of rotating quantum states population for 14N 2 and 15N 2 molecules interacting with a train of picoseconds laser pulses with different train period and intensity. We show for some train periods there are resonances of population transfer from low rotating quantum states of investigated molecules to high states. We study these resonances for various laser ﬁeld intensities and pulses train periods. We note that in resonance case the parameters of laser ﬁeld are different for 14N 2 and 15N 2 molecules. Obtained results indicate on the possibility of molecules rotating states selective exitation by ultrashort laser pulses train. Our numerical results are in agreement with results of experimental studies [Phys. Rev. Lett., 2012, vol. 109, 043003].

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(2):221-240

Lévy d'Alambertians and their application in the quantum theory

#### Abstract

The Lévy d'Alambertian is the natural analogue of the well-known Lévy Laplacian. The aim of the paper is the following. We study the relationship between different deﬁnitions of the Lévy d'Alambertian and the relationship between the Lévy d'Alambertian and the QCD equations (the Yang-Mills-Dirac equations). There are two different deﬁnitions of the classical Lévy d'Alambertian. One can deﬁne the Lévy d'Alambertian as an integral funce tional given by the second derivative or deﬁne it using the Cesaro means of the directional derivatives along the elements of some orthonormal basis. Using the weakly uniformly dense bases we prove the equivalence of these two deﬁnitions. We introduce the family of the nonclassical Lévy d'Alambertians using the family of the nonclassical L´vy Laplacians as a model. Any element of this family is associated with the linear operator on the linear span of the orthonormal basis. The classical Lévy d’Alambertian is an element of this family associated with the identity operator. We can describe the connection between the Lévy d'Alambertians and the gauge ﬁelds using the classical Lévy d'Alambertian or another nonclassical Lévy d'Alambertian speciﬁed in this paper. We study the relationship between this nonclassical Lévy d'Alambertian and the Yang-Mills equations with a source and obtain the system of inﬁnite dimensional differential equations which is equivalent to the QCD equations.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(2):241-258

Description of radiative decays of $V \to P \gamma^{*}$ in different forms of Poincaré-invariant quantum mechanics

#### Abstract

The description of the radiative decays of $V \to P \gamma^*$ in different forms Poincaré invariant quantum mechanics (PIQM) is considered. To construct the matrix element of the electromagnetic current we use the non-diagonal parametrization procedure. The obtained matrix element of the current satisfies the conditions of the Lorentz covariance and conservation. To illustrate this approach in a modified relativistic impulse approximation the description of the radiative transition $\rho \to \pi \gamma^{*}$ is performed. An analytic expression for the transition form factor $F_{\pi \rho}(Q^2)$, matching all forms PIQM, is obtained. Numerical calculations of the transition form factor are made with different model wave functions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(2):259-269

Hyperfine structure of muonic lithium ions

#### Abstract

On the basis of perturbation theory in fine structure constant $\alpha$ and the ratio of electron to muon masses we calculate recoil corrections of order $\alpha^4 (M_e/M_\mu)$, $\alpha^4 (M_e/M_\mu)^2\ln(M_e/M_\mu)$, $\alpha^4 (M_e/M_\mu)^2$, $\alpha^5(m_e/m_\mu)\ln(m_e/m_\mu)$ to hyperfine splitting of the ground state in muonic lithium ions $(\mu\ e\ ^6_3\mathrm{Li})^+$ and $(\mu\ e\ ^7_3\mathrm{Li})^+$. We obtain total results for the ground state small hyperfine splittings in $(\mu\ e\ ^6_3\mathrm{Li})^+$ $\Delta\nu_1=14153.03$~MHz and $\Delta\nu_2=21571.26$~MHz and in $(\mu\ e\ ^7_3\mathrm{Li})^+$ $\Delta\nu_1=13991.97$~MHz and $\Delta\nu_2=21735.03$~MHz which can be considered as a reliable estimate for a comparison with future experimental data.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(2):270-282

De la Vallée Poussin problem in the kernel of the convolution operator on the half-plane

#### Abstract

We consider the multipoint de la Vallée Poussin (interpolational) problem in the half-plane $D$, $D=\{z \, :\, \mathop{\mathrm{Re}} z<\alpha,$ $ \alpha>0\}$. Let $\psi(z)\in H(D)$; $\mu_1$, $\mu_2$,~$\ldots \in D$ be the positive zero points of this function and let the boundary of domain $D$ contain their limit. Also, we assume that $\mu_k$ is of $s_k$ multiplicity, $k=1, 2, \dots$. Let us set $M_{\varphi}$ an operator of convolution with the characteristic function $\varphi(z)$. Taking an arbitrary sequence $a_{kj},$ $j=0, 1, \ldots, s_k-1$ we should ask: is there a function $u(z) \in \mathop{\mathrm{Ker}}M_\varphi$ that provides the relation $u^{(j)}(\mu_{k})=a_{kj},$ $j=0, 1,\dots,s_k-1$? We assume the operator characteristic function to be of completely regular growth. The solvability conditions for the multipoint de la Vallée Poussin problem in the half-plain and in the bounded convex domains are obtained.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(2):283-292

Scattering of vortices in abelian higgs models on compact riemann surfaces

#### Abstract

Abelian Higgs models on Riemann surfaces are natural analogues of the (2 + 1)-dimensional Abelian Higgs model on the plane. The latter model arises in theory of superconductivity. For this model the following result was previously obtained: if two vortices (zeros of the Higgs ﬁeld) move slowly, then after the head-on collision they scatter under the right angle, and if N vortices collide, then after the symmetric head-on collision they scatter on the angle π/N . In the critical case (when the parameter of the model is equal to 1) these results can be obtained with the help of so-called adiabatic principle. This principle allows to consider geodesics of so-called kinetic metric (metric that is given by kinetic energy functional) on the moduli space of static solutions as approximations to dynamical solutions of the model with small kinetic energy. Recently, the adiabatic principle was rigorously justiﬁed in the (2 + 1)-dimensional Abelian Higgs model on the plane in the critical case. Although the metric can not be written in explicit form, one can prove that required geodesics (describing the π/N scattering) exist, using smoothness of the metric in coordinates that are given by symmetric functions on positions of vortices and symmetry properties of the kinetic metric. A local analogue of the result on π/N scattering in (2+1)-dimensional Abelian Higgs model on the plane can be deduced only from smoothness property of the kinetic metric. One can suppose that this local version of the result on π/N scattering can be generalized to Abelian Higgs models on Riemann surfaces. It is proved in this paper that one can ﬁnd geodesics of the kinetic metric that describe local π/N scattering after the symmetric collision in models on Riemann surfaces, using the fact that the kinetic metric is smooth in symmetric coordinates in the neihbourhood of a point of vortex collision. This smoothness property is established in the case of compact Riemann surfaces. With the help of adiabatic principle one could obtain local π/N scattering after the symmetric collision for dynamical models on compact Riemann surfaces. Unfortunately, the adiabatic principle in models on compact Riemann surfaces needs the proof yet, until now it is only a heuristic statement.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(2):293-310

Fluctuations of a beam with clamped ends

#### Abstract

In this paper we study the initial problem for the equation of a beam with clamped ends. Uniqueness, existence and stability theorems are proved for the problem in the classes of regular and generalized solutions. Solution of the initial-boundary value problem is constructed in the form of a series in the system of eigenfunctions of one-dimensional spectral problem. We found the spectral problem eigenvalues as roots of the transcendental equation and the corresponding system of eigenfunctions. It is shown that the system of eigenfunctions is orthogonal and complete in L 2. On the basis of the completeness of the eigenfunctions the uniqueness theorem for the initial-boundary value problem for the equation of the beam is obtained. The generalized solution is deﬁned as the limit of a sequence of regular solutions of the mean-square norm on the space variable.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(2):311-324

On frame indifferent lagrangians of micropolar thermoelastic continuum

#### Abstract

A non-linear mathematical model of type-II thermoelastic continuum with ﬁne microstructure is developed. The model is described in terms of 4covariant ﬁeld theoretical formalism attributed to ﬁeld theories of continuum mechanics. Fine microstructure is introduced by d-vectors and tensors playing role of extra ﬁeld variables. A Lagrangian density for type-II thermoelastic continuum with ﬁne microstructure is proposed and the least action principle is formulated. Virtual microstructural inertia is added to the considered action density. It is also valid for the thermal inertia. Corresponding 4-covariant ﬁeld equations of type-II thermoelasticity are obtained. Constitutive equations of type-II microstructural thermoelasticity are discussed. Following the usual procedure for type-II micropolar thermoelastic Lagrangians functionally independent rotationally invariant arguments are obtained. Those are proved to form a complete set. Objective forms of the Lagrangians satisfying the frame indifference principle are given. Those are derived by using extrastrain vectors and tensors.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(2):325-340

Overall stability of compressed compound bars in variable cross section

#### Abstract

The article provides the solution to the resistance problem of centrally compressed compound bars of variable section with power-law hardening. The assumed model of S. Timoshenko's theory is valid for calculating and analyzing the general resistance of tower-type bar systems (towers, masts, trestle supports) under certain conditions of stiffening behavior. Unlike the traditional way, the plastic design of lateral shear is made on the basis of independent equilibrium equations. The article describes the condition under which the traditional approach accepted in the technical rod theory and based on the inner forces correlation is valid. Boundary value problem is formulated on the basis of the resolving quadratic equation (traditional approach) and the equation of fourth order (more general, suggested in the paper, adequate deﬁnition). For this purpose the technique associated with the increase of the resolving equation’s degree is used. In the ﬁrst case it is possible to examine only symmetric forms of resistance loss. In the second case both symmetric and asymmetric forms of resistance loss are possible to be examined. Transcendental equation of resistance for different cases of bar's ﬁxing is obtained. The coeffcients of the given length are analyzed depending upon the ways of ﬁxing the end sections. The article points out that unlike the situation with the bars of solid cross-section it is necessary to take into account the shear strain of the grid in the compound bars of variable stiffness while examining their general buckling resistance.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(2):341-357

Mixed-mode loading of the structural elements with defect

#### Abstract

In the article the problem of determining the stress-strain state near the mixed-mode crack tip in a power-law material under plane stress conditions is considered. The eigenfunction method is used for the mixed-mode crack tip problem. It is shown that the eigenfunction expansion method results in the nonlinear eigenvalue problem. The numeric solution of the nonlinear eigenvalue problem formulated is obtained. The power of the distance from the crack tip is the eigenvalue of the nonlinear eigenvalue problem considered whereas the angular distributions of the stress components are the eigenfunctions. The new eigenvalues different from the eigenvalues of the Hutchinson-Rice-Rosengren are found. It is shown that the new asymptotic solution can be interpreted as the self-similar intermediate asymptotics of the stress ﬁeld in the vicinity of the crack tip at distances which are very small compared to the crack length or the size of the specimen and at distances which are large compared to the length of the completely damaged zone. The developed method allows us to construct the geometry of the completely damaged zone in vicinity of the crack tip.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(2):358-381

Mathematical models of nonlinear longitudinal-cross oscillations of object with moving borders

#### Abstract

The nonlinear formulation of problems for describing longitudinal-cross oscillations of objects with moving borders is noted. These mathematical models consist of a system of two nonlinear partial differential equations with the higher time derivative of the second order and the fourth-order by the spatial variable. The nonlinear boundary conditions on moving boundary have a higher time derivative of the second order and the third-order by the spatial variable. The geometric nonlinearity, visco-elasticity, the ﬂexural stiffness of the oscillating object and the elasticity of the substrate of object are taken into account. Boundary conditions in the case of energy exchange between the parts of the object on the left and right of the moving boundary are given. The moving boundary has got a joined mass. The elastic nature of borders joining is considered. The longitudinal-cross oscillations of objects with moving borders of high intensity can be described by the resulting differential model. The Hamilton’s variational principle is used in the formulation of the problem.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(2):382-397

Orthogonal Franklin system and orthogonal system of finite functions in numerical methods of boundary problems solving

#### Abstract

Possibilities of classical trigonometric Fourier series are substantially limited in 2-D and 3-D boundary value problems. Boundary conditions of such problems for areas with curvilinear boundaries often fails when using the classical Fourier series. The solution of this problem is the use of orthogonal ﬁnite functions. However, orthogonal Haar basis functions are not continuous. The orthogonal Daubechies wavelets have compact supports, but is not written in analytical form and have low smoothness. Continuous ﬁnite Schauder-Faber functions are not orthogonal. Orthogonal Franklin continuous functions are not ﬁnite. The connection of the orthogonal Franklin functions with a sequence of grid groups of piecewise linear orthogonal ﬁnite basis functions (OFF) is established here. The Fourier-OFF series on the basis of such continuous OFF is formed. Such series allows to execute boundary conditions of Dirichlet’s type on curvilinear boundaries in integral performances of boundary value problems. A similar problem is connected with a satisfaction of Neumann boundary conditions and also is eliminated in the integral mixed performances of boundary value problems. Fourier-OFF series increases the eﬀectiveness of mixed numerical methods for boundary value problems solving.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(2):398-404