Semigroup of positive maps for qudit states and entanglement in tomographic probability representation

Stochastic and bistochastic matrices providing positive maps for spin states (for qudits) are shown to form semigroups with dense intersection with the Lie groups IGL(n,R) and GL(n,R) respectively. The density matrix of a qudit state is shown to be described by a spin tomogram determined by an orbit of the bistochastic semigroup acting on a simplex. A class of positive maps acting transitively on quantum states is introduced by relating stochastic and quantum stochastic maps in the tomographic setting. Finally, the entangled states of two qubits and Bell inequalities are given in the framework of the tomographic probability representation using the stochastic semigroup properties.     

Spin-0 and spin-1/2 particles in a constant scalar-curvature background

We study the Klein-Gordon and Dirac equations in the presence of a background metric ds²=−dt²+dx²+e^−2gx(dy²+dz²) in a semi-infinite lab (x>0). This metric has a constant scalar-curvature R=6g² and is produced by a perfect fluid with equation of state p=−ρ/3. The eigenfunctions of spin-0 and spin-1/2 particles are obtained exactly, and the quantized energy eigenvalues are compared. It is shown that both of these particles must have nonzero transverse momentum in this background. We show that there is a minimum energy E²_min=m²c⁴+g²c²?² for bosons (E_KG>E_min), while the fermions have no specific ground state (E_Dirac>mc²).     

Unified path integral treatment for generalized Hulthén and Woods-Saxon potentials

A rigorous path integral discussion of the s states for a diatomic molecule potential with varying shape, which generalizes the Hulthén and the Woods-Saxon potentials, is presented. A closed form of the Green’s function is obtained for different shapes of this potential. For λ ? 1 and (1/η)lnλ<r<∞, the energy spectrum and the normalized wave functions of the bound states are derived. When the deformation parameter λ is 0 < λ < 1 or λ < 0, it is found that the quantization conditions are transcendental equations that require numerical solutions. The special cases corresponding to a screened potential (λ = 1), the deformed Woods-Saxon potential (λ = q e^ηR), and the Morse potential (λ = 0) are likewise treated.     

Vortex structure of Gross-Pitaevskii model

The vortex line of the Gross-Pitaevskii model is studied. The kinetic helicity of the vortex is discussed, and vortex structure is classified by the Hopf index, linking number in geometry. A mechanism of generation and annihilation of vortex lines is given by the method of phase singularity theory. The dynamic behavior of the vortex at the critical points is discussed in detail, and three kinds of length approximation relations at the neighborhood of a critical point are given: l ∝ (t − t^∗)^1/2, l ∝ t − t^∗, l = const.     

First-order intertwining operators and position-dependent mass Schrödinger equations in d dimensions

The problem of d-dimensional Schrödinger equations with a position-dependent mass is analyzed in the framework of first-order intertwining operators. With the pair (H, H₁) of intertwined Hamiltonians one can associate another pair of second-order partial differential operators (R, R₁), related to the same intertwining operator and such that H (resp. H₁) commutes with R (resp. R₁). This property is interpreted in superalgebraic terms in the context of supersymmetric quantum mechanics (SUSYQM). In the two-dimensional case, a solution to the resulting system of partial differential equations is obtained and used to build a physically relevant model depicting a particle moving in a semi-infinite layer. Such a model is solved by employing either the commutativity of H with some second-order partial differential operator L and the resulting separability of the Schrödinger equation or that of H and R together with SUSYQM and shape-invariance techniques. The relation between both approaches is also studied.     

Algebraic treatments of the problems of the spin-1/2 particles in the one- and two-dimensional geometry: A systematic study

We consider solutions of the 2 × 2 matrix Hamiltonians of the physical systems within the context of the su (2) and su (1, 1) Lie algebras. Our technique is relatively simple when compared with those of others and treats those Hamiltonians which can be treated in a unified framework of the Sp (4, R) algebra. The systematic study presented here reproduces a number of earlier results in a natural way as well as leads to a novel finding. Possible generalizations of the method are also suggested.     

Collective quantum decoherence induced by qubit-electron interaction

We investigate a model of quantum register composed of N qubits coupling with itinerant electrons by adopting the Born-Markov master equation. Decoherence induced by this coupling is studied for various initial states. By solving the master equation for N=4 with the numerical integration, we obtain time evolution of fidelity and linear entropy of the register. The decoherence rate of this model is proportional to ²|J^′| with J^′ being the exchange coupling strength of electrons and qubits. We also investigate the decoherence free subspace which provides a possible routine of applications in quantum computation.     

New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation

We analyze the time evolution of mixed state ρ₀ in a dissipative channel, characteristic of a decay constant κ, by virtue of the elegant properties of entangled state representation 〈η|. We find that the matrix element of the mixed state ρ(t) at time t in 〈η| representation is proportional to that of the initial ρ₀ in the decayed entangled state 〈ηe^-κt| representation, accompanying with a Gaussian damping factor . Thus we have a new insight about the nature of the dissipative process.     

Regularization in the J-matrix method of scattering revisited

We present an alternative, but equivalent, approach to the regularization of the reference problem in the J-matrix method of scattering. After identifying the regular solution of the reference wave equation with the “sine-like” solution in the J-matrix approach we proceed by direct integration   to find the expansion coefficients in an L²$L^2$ basis set that ensures a tridiagonal representation of the reference Hamiltonian. A differential equation in the energy is then deduced for these coefficients. The second independent solution of this equation, called the “cosine-like” solution, is derived by requiring it to pertain to the L²$L^2$ space. These requirements lead to solutions that are exactly identical to those obtained in the classical J-matrix approach. We find the present approach to be more direct and transparent than the classical differential approach of the J-matrix method.     

A relativistic wave equation for the Skyrmion

We propose a relativistically invariant wave equation for the Skyrme soliton. It is a differential equation on the space R^1,3×S³ which is invariant under the Lorentz group and isospin. The internal variable valued in SU(2)≡S³ describes the orientation of the soliton. The mass of a particle of spin and isospin both equal to is predicted to be which agrees with the known spectrum for low angular momentum. The iso-scalar magnetic moment is predicted to be , where Σ is the spin.     

Singularities of Berry connections inhibit the accuracy of the adiabatic approximation

Adiabatic approximation for quantum evolution is investigated addressing its dependence on the Berry connections that are functions of a slowly-varying parameter R  . When the Berry connections have singularities of type 1/Rσ$1/R^{\sigma }$ with σ<1$\sigma <1$, the adiabatic fidelity converges to unit according to a power-law; When the singularity index σ becomes larger than one, adiabatic approximation breaks down. Two-level models are used to substantiate our theory.     

Determination of the central density on the basis of its moments

The value of the central density is of key importance for annihilation processes. For the ground state we discuss its determination from the moments of the ground state density. We first review the way of reaching the moments from the spectrum. In particular we show how to get the lowest moments in D = 3, namely 〈r⁻²〉 and 〈r⁻¹〉 from the series expansion of the Laplace transform of the density. We then recall a method to obtain the central density based on the Stieltjes moment problem. If the number of known moments is finite, this technique yields a lower bound. We investigate the possibilities to estimate the accuracy of the bound and the corresponding asymptotic value. An application to the muonic ²⁰⁸Pb atom is presented.     

Operator equations and Moyal products–metrics in quasi-Hermitian quantum mechanics

The Moyal product is used to cast the equation for the metric of a non-Hermitian Hamiltonian in the form of a differential equation. For Hamiltonians of the form p²+V(ix)$p^2+V(ix)$ with V polynomial this is an exact equation. Solving this equation in perturbation theory recovers known results. Explicit criteria for the hermiticity and positive definiteness of the metric are formulated on the functional level.     

A quantum exactly solvable non-linear oscillator with quasi-harmonic behaviour

The quantum version of a non-linear oscillator, previously analyzed at the classical level, is studied. This is a problem of quantization of a system with position-dependent mass of the form m = (1 + λx²)⁻¹ and with a λ-dependent non-polynomial rational potential. This λ-dependent system can be considered as a deformation of the harmonic oscillator in the sense that for λ → 0 all the characteristics of the linear oscillator are recovered. First, the λ-dependent Schrödinger equation is exactly solved as a Sturm-Liouville problem, and the λ-dependent eigenenergies and eigenfunctions are obtained for both λ > 0 and λ < 0. The λ-dependent wave functions appear as related with a family of orthogonal polynomials that can be considered as λ-deformations of the standard Hermite polynomials. In the second part, the λ-dependent Schrödinger equation is solved by using the Schrödinger factorization method, the theory of intertwined Hamiltonians, and the property of shape invariance as an approach. Finally, the new family of orthogonal polynomials is studied. We prove the existence of a λ-dependent Rodrigues formula, a generating function and λ-dependent recursion relations between polynomials of different orders.     

Diffraction in the semiclassical approximation to Feynman's path integral representation of the Green function

We derive the semiclassical approximation to Feynman's path integral representation of the energy Green function of a massless particle in the shadow region of an ideal obstacle in a medium. The wavelength of the particle is assumed to be comparable to or smaller than any relevant length of the problem. Classical paths with extremal length partially creep along the obstacle and their fluctuations are subject to non-holonomic constraints. If the medium is a vacuum, the asymptotic contribution from a single classical path of overall length L to the energy Green function at energy E is that of a non-relativistic particle of mass E/c² moving in the two-dimensional space orthogonal to the classical path for a time τ=L/c. Dirichlet boundary conditions at the surface of the obstacle constrain the motion of the particle to the exterior half-space and result in an effective time-dependent but spatially constant force that is inversely proportional to the radius of curvature of the classical path. We relate the diffractive, classically forbidden motion in the “creeping” case to the classically allowed motion in the “whispering gallery” case by analytic continuation in the curvature of the classical path. The non-holonomic constraint implies that the surface of the obstacle becomes a zero-dimensional caustic of the particle's motion. We solve this problem for extremal rays with piecewise constant curvature and provide uniform asymptotic expressions that are approximately valid in the penumbra as well as in the deep shadow of a sphere.     

Approach to the quantum evolution for underdamped, critically damped, and overdamped driven harmonic oscillators using unitary transformation

Exact solution of the Schrödinger equation is derived for underdamped, critically damped, and overdamped harmonic oscillators with a driving force. A unitary operator transforming Hamiltonian into a simple form is introduced. The transformed Hamiltonian, represented in terms of a modified frequency ω, is identical with the Hamiltonian of the standard harmonic oscillator for the underdamped oscillator, with the Hamiltonian of a free particle for the critically damped oscillator, and with the Hamiltonian of a system with a harmonic parabolic potential for the overdamped oscillator. The eigenvalues of underdamped oscillator are discrete while those of the critically damped and the overdamped oscillators are continuous.     

Eigenvalues, eigenfunctions, and surface states in finite periodic systems

Using a simple approach that requires neither the Bloch functions nor the reciprocal lattice, new, compact, and rigorous analytical formulas are derived for an accurate evaluation of resonant energies, resonant states, energy eigenvalues and eigenfunctions of open and bounded n-cell periodic systems with arbitrary 1D potential shapes, provided the single cell transfer matrix is given. These formulas are applied to obtain the energy spectra and wave functions of a number of simple but representative open and bounded superlattices. We solve the fine structure in bands and exhibit unambiguously that the true eigenfunctions do no not fulfill the periodicity property |Ψ_μ,ν (z + l_c)|² = |Ψ_μ,ν (z)|², with l_c the single cell length. We show that the well known surface states and surface energy levels come out naturally. We analyze the surface repulsion effect and calculate exactly the surface energy levels for different potential discontinuities an the ends.     

SUSY-hierarchy of one-dimensional reflectionless potentials

A class of one-dimensional reflectionless potentials is studied. It is found that all possible types of the reflectionless potentials can be combined into one SUSY-hierarchy with a constant potential. An approach for determination of a general form of the reflectionless potential on the basis of construction of such a hierarchy by the recurrent method is proposed. A general integral form of interdependence between superpotentials with neighboring numbers of this hierarchy, opening a possibility to find new reflectionless potentials, is found and has a simple analytical view. It is supposed that any possible type of the reflectionless potential can be expressed through finite number of elementary functions (unlike some presentations of the reflectionless potentials, which are constructed on the basis of soliton solutions or are shape invariant in one or many steps with involving scaling of parameters, and are expressed through series). An analysis of absolute transparency existence for the potential which has the inverse power dependence on space coordinate (and here tunneling is possible), i.e., which has the form V (x) = ± α/|x−x₀|ⁿ (where α and x₀ are constants, n is natural number), is fulfilled. It is shown that such a potential can be reflectionless at n = 2 only. A SUSY-hierarchy of the inverse power reflectionless potentials is constructed. Isospectral expansions of this hierarchy are analyzed.     

Tensor representation in teleportation and controlled teleportation

We propose the tensor representation of teleportation and controlled teleportation. By using this representation, it is easy to describe the process of teleporting an unknown N-qubit state via a genuine 2N-qubit channel, and to find the necessary and sufficient condition of realizing a successful teleportation (which is determined by the measurement matrix T^α and the quantum channel parameter matrix X). For controlled teleportation, if composing tensor representation with graph, one can easily design any kind of controlled teleportation. As examples, we give a scheme of symmetrically controlled teleportation of two-qubit states and a scheme of representative network controlled of three-qubit states. This method can also be generalized to the controlled teleportation of N-qubit states.