Nonrelativistic Quantum Mechanics with Fundamental Environment

Spontaneous transitions between bound states of an atomic system, “Lamb Shift” of energy levels and many other phenomena in real nonrelativistic quantum systems are connected within the influence of the quantum vacuum fluctuations (fundamental environment (FE)) which are impossible to consider in the limits of standard quantum-mechanical approaches. The joint system “quantum system (QS) + FE” is described in the framework of the stochastic differential equation (SDE) of Langevin-Schr?dinger (L-Sch) type, and is defined on the extended space R ³ ⊗ R _{ξ}, where R ³ and R _{ξ} are the Euclidean and functional spaces, respectively. The density matrix for single QS in FE is defined. The entropy of QS entangled with FE is defined and investigated in detail. It is proved that as a result of interaction of QS with environment there arise structures of various topologies which are a new quantum property of the system.     

Exactly constructing model of quantum mechanics with random environment

Dissipation and decoherence, interaction with the random media, continuous measurements and many other complicated problems of open quantum systems are a result of interaction of quantum system with the random environment. These problems mathematically are described in terms of complex probabilistic processes (CPP). Note that CPP satisfies the stochastic differential equation (SDE) of Langevin—Schrödinger(L—Sch)type, and is defined on the extended space R¹ ? R_{γ}, where R¹ and R_{γ} are the Euclidean and the functional spaces, correspondingly. For simplicity, the model of 1D quantum harmonic oscillator (QHO) with the stochastic environment is considered. On the basis of orthogonal CPP, the method of stochastic density matrix (SDM) is developed. By S DM method, the thermodynamical potentials, such as the nonequilibrium entropy and the energy of the “ground state” are constructed in a closed form. The expressions for uncertain relations and Wigner function depending on interaction’s constant between 1D QHO and the environment are obtained.     

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.     

On the exact mean-field description of continuous quantum systems in equilibrium

The thermodynamic limit of free energy density is investigated for quantum systems of n particles obeying Boltzmann, Fermi and Bose statistics, interacting via spin-independent 2-body bounded separable potentials and confined to a bounded region Λ ? R^v. The technique used exploits the Feynman-Kac theorem in finite volume and the saddle-point method of Tindemans and Capel. It is shown that the limiting free energy density of such systems is equal to that of a system of noninteracting particles subject to a mean field which is equal to the averaged 2-body interaction. The equations for the mean field of n particles obeying Boltzmann, Fermi or Bose statistics represent self-consistent field problems and their forms comply with the well-known theorems on mean occupation numbers of single-particle eigenstates of ideal quantum gases at inverse temperature β.     

On the Distribution of the Wave Function for Systems in Thermal Equilibrium

For a quantum system, a density matrix ρ that is not pure can arise, via averaging, from a distribution μ of its wave function, a normalized vector belonging to its Hilbert space ?. While ρ itself does not determine a unique μ, additional facts, such as that the system has come to thermal equilibrium, might. It is thus not unreasonable to ask, which μ, if any, corresponds to a given thermodynamic ensemble? To answer this question we construct, for any given density matrix ρ, a natural measure on the unit sphere in ?, denoted GAP(ρ). We do this using a suitable projection of the Gaussian measure on ? with covariance ρ. We establish some nice properties of GAP(ρ) and show that this measure arises naturally when considering macroscopic systems. In particular, we argue that it is the most appropriate choice for systems in thermal equilibrium, described by the canonical ensemble density matrix ρ_β = (1/Z) exp (?β H). GAP(ρ) may also be relevant to quantum chaos and to the stochastic evolution of open quantum systems, where distributions on ? are often used.     

Propagation of excitation in long 1D chains: Transition from regular quantum dynamics to stochastic dynamics

The quantum dynamics problem for a 1D chain consisting of 2N + 1 sites (N ? 1) with the interaction of nearest neighbors and an impurity site at the middle differing in energy and in coupling constant from the sites of the remaining chain is solved analytically. The initial excitation of the impurity is accompanied by the propagation of excitation over the chain sites and with the emergence of Loschmidt echo (partial restoration of the impurity site population) in the recurrence cycles with a period proportional to N. The echo consists of the main (most intense) component modulated by damped oscillations. The intensity of oscillations increases with increasing cycle number and matrix element C of the interaction of the impurity site n = 0 with sites n = ±1 (0 < C ≤ 1; for the remaining neighboring sites, the matrix element is equal to unity). Mixing of the components of echo from neighboring cycles induces a transition from the regular to stochastic evolution. In the regular evolution region, the wave packet propagates over the chain at a nearly constant group velocity, embracing a number of sites varying periodically with time. In the stochastic regime, the excitation is distributed over a number of sites close to 2N, with the populations varying irregularly with time. The model explains qualitatively the experimental data on ballistic propagation of the vibrational energy in linear chains of CH₂ fragments and predicts the possibility of a nondissipative energy transfer between reaction centers associated with such chains.     

The number process as low density limit of Hamiltonian models

We study a nonrelativistic quantum system coupled, via a quadratic interaction [cf. formula (1.10) below], to a free Boson gas in the Fock state. We prove that, in the low density limit (z ²=fugacity0), the matrix elements of the wave operator of the system at timet/z ² in the collective coherent vectors converge to the matrix elements, in suitable coherent vectors of the quantum Brownian motion process, of a unitary Markovian cocycle satisfying a quantum stochastic differential equation driven by some pure number process (i.e. no quantum diffusion part and only the quantum analogue of the purely discontinuous, or jump, processes). This proves that the number (or quantum Poisson) processes, introduced by Hudson and Parthasarathy and studied by Frigerio and Maassen, arise effectively as conjectured by the latter two authors as low density limits of Hamiltonian models.     

Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles

This paper is devoted to the rigorous proof of the universality conjecture of random matrix theory, according to which the limiting eigenvalue statistics ofn×n random matrices within spectral intervals ofO(n ^–1) is determined by the type of matrix (real symmetric, Hermitian, or quaternion real) and by the density of states. We prove this conjecture for a certain class of the Hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function (the potential in the quantum field theory) satisfying some regularity conditions.     

Dynamics of collective decoherence and thermalization

We analyze the dynamics of N interacting spins (quantum register) collectively coupled to a thermal environment. Each spin experiences the same environment interaction, consisting of an energy conserving and an energy exchange part.We find the decay rates of the reduced density matrix elements in the energy basis. We show that if the spins do not interact among each other, then the fastest decay rates of off-diagonal matrix elements induced by the energy conserving interaction is of order N², while that one induced by the energy exchange interaction is of the order N only. Moreover, the diagonal matrix elements approach their limiting values at a rate independent of N. For a general spin system the decay rates depend in a rather complicated (but explicit) way on the size N and the interaction between the spins.Our method is based on a dynamical quantum resonance theory valid for small, fixed values of the couplings. We do not make Markov-, Born- or weak coupling (van Hove) approximations.     

Quantum Image Encryption Based on Iterative Framework of Frequency-Spatial Domain Transforms

A novel quantum image encryption and decryption algorithm based on iteration framework of frequency-spatial domain transforms is proposed. In this paper, the images are represented in the flexible representation for quantum images (FRQI). Previous quantum image encryption algorithms are realized by spatial domain transform to scramble the position information of original images and frequency domain transform to encode the color information of images. But there are some problems such as the periodicity of spatial domain transform, which will make it easy to recover the original images. Hence, we present the iterative framework of frequency-spatial domain transforms. Based on the iterative framework, the novel encryption algorithm uses Fibonacci transform and geometric transform for many times to scramble the position information of the original images and double random-phase encoding to encode the color information of the images. The encryption keys include the iterative time t of the Fibonacci transform, the iterative time l of the geometric transform, the geometric transform matrix G i which is n × n matrix, the classical binary sequences K (\(k_{0}k_{1}{\ldots } k_{2^{2n}-1}\)) and \(D(d_{0}d_{1}{\ldots } d_{2^{2n}-1}\)). Here the key space of Fibonacci transform and geometric transform are both estimated to be 2²⁶. The key space of binary sequences is (2 ^n×n ) × (2 ^n×n ). Then the key space of the entire algorithm is about \(2^{2{n^{2}}+52}\). Since all quantum operations are invertible, the quantum image decryption algorithm is the inverse of the encryption algorithm. The results of numerical simulation and analysis indicate that the proposed algorithm has high security and high sensitivity.     

Transition energy and dipole oscillator strength of 1s23d-1s2 nf transitions for Sc18+ ion

The transition energies of the 1s²3d-1s² nf (4?n?9) transitions and fine structure splittings of 1s² nf (n?9) states for Sc¹⁸⁺ ion are calculated with the full-core plus correlation method. The quantum defect of 1s² nf series is determined by the single-channel quantum defect theory. The energies of any highly excited states with n?10 for this series can be reliably predicted using the quantum defect as function of energy. Three alternative forms of the dipole oscillator strengths for the 1s²3d-1s² nf (n?9) transitions of Sc¹⁸⁺ ion are calculated with the transition energies and wave functions obtained above. Combining the quantum defect theory with the discrete oscillator strengths, the discrete oscillator strengths for 1s²3d-1s² nf (n > 9) transitions and the oscillator strengths densities corresponding to the bound-free transitions are obtained.     

Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion

The perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory. We show that the algebra of interacting fields is additive, i.e. fully determined by its subalgebras associated to arbitrary small subregions of Minkowski space. We also give an algebraic formulation of the loop expansion by introducing a projective system ?^{( n )} of observables “up to n loops”, where ?⁽⁰⁾ is the Poisson algebra of the classical field theory. Finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of Green's functions. Received: 9 February 2000 / Accepted: 21 March 2000     

A Stochastic Mechanics Based on Bohm‧s Theory and its Connection with Quantum Mechanics

We construct a stochastic mechanics by replacing Bohm‧s first-order ordinary differential equation of motion with a stochastic differential equation where the stochastic process is defined by the set of Bohmian momentum time histories from an ensemble of particles. We show that, if the stochastic process is a purely random process with n-th order joint probability density in the form of products of delta functions, then the stochastic mechanics is equivalent to quantum mechanics in the sense that the former yields the same position probability density as the latter. However, for a particular non-purely random process, we show that the stochastic mechanics is not equivalent to quantum mechanics. Whether the equivalence between the stochastic mechanics and quantum mechanics holds for all purely random processes but breaks down for all non-purely random processes remains an open question.     

Nonexponential decay in the quantum dynamics of nanosystems

The quantum dynamical problem is solved for a system coupled to an equidistant-spectrum bath with the energy difference Ω between the neighboring levels n and n + 1 and the coupling matrix elements C _n ² = C ²(1 + Δ^?2 n ²)^?1 constraining the energy interval comprising the bath states interacting with the system. The evolution in the strong-coupling limit is determined by two parameters, Γ = πC ²/Ω ? 1 and α = Γ/Δ. If α ≠ 0, then the decrease in the population in the initial cycle with a period of 2π/Ω is not exponential and the effective rate constant increases with time. The results qualitatively explain the appearance of nonexponential relaxation regimes for a dense-spectrum nanosystem and predict the possibility of the multiple recovery of the initial-state population.     

Universal two-dimensional kinetics of the populations of Rydberg atoms in plasmas

Universal kinetics has been developed for calculating the two-dimensional populations (in the space of principal n and orbital l quantum numbers) of Rydberg atomic states in plasmas. It has been shown that the direct population of the atomic states by three-body and radiative recombination sources should be taken into account, because the radiative cascade is of quantum character. The subsequent two-dimensional radiation-collision cascade is constructed in the framework of the classical approach. The developed model makes it possible to obtain the populations in the form of universal functions of temperature and density. The numerical calculations of the populations of highly excited hydrogen states (n ∝ 100) in low-density plasmas (10³–10¹³ cm^?3) at moderate temperatures (1 eV) indicate a significantly nonequilibrium population both in n and in l, which is important for the diagnostics of astrophysical and laboratory plasmas.     

Variational calculations of asymmetric nuclear matter

We report on variational calculations of the energy E(ρ, β) of asymmetric nuclear matter having ? = ?_n + ?_p = 0.05 to 0.35 fm^?3, and β = (?_n ? ?_p/g9 = 0 to 1. The nuclear h used in this work consists of a realistic two-nucleon interaction, called v₁₄, that fits the available nucleon-nucleon scattering data up to 425 MeV, and a phenomenological three nucleon interaction adjusted to reproduce the empirical properties of symmetric nuclear matter. The variational many-body theory of symmetric nuclear matter is extended to treat matter with neutron excess. Numerical and analytic studies of the β-dependence of various contributions to the nuclear matter energy show that at ? < 0.35 fm^?3 the β⁴ terms are very small, and that the interaction energy EI(ρ, β) defined as E(ρ, β) ? T_F(ρ, β), where T_F is the Fermi-gas energy, is well approximated by EI₀(?) + β²EI₂(ρ). The calculated symmetry energy at equilibrium density is 30 MeV and it increases from 15 to 38 MeV as ? increases from 0.05 to 0.35 fm^?3.     

Transfer of a weakly bound electron in collisions of Rydberg atoms with neutral particles. II. Ion-pair formation and resonant quenching of the Rb(nl) and Ne(nl) States by Ca, Sr, and Ba atoms

Electron-transfer processes are studied in thermal collisions of Rydberg atoms with alkaline-earth Ca(4s ²), Sr(5s ²), and Ba(6s ²) atoms capable of forming negative ions with a weakly bound outermost p-electron. We consider the ion-pair formation and resonant quenching of highly excited atomic states caused by transitions between Rydberg covalent and ionic terms of a quasi-molecule produced in collisions of particles. The contributions of these reaction channels to the total depopulation cross section of Rydberg states of Rb(nl) and Ne(nl) atoms as functions of the principal quantum number n are compared for selectively excited nl-levels with l ? n and for states with large orbital quantum numbers l = n ? 1, n ? 2. It is shown that the contribution from resonant quenching dominates at small values of n, and the ion-pair formation process begins to dominate with increasing n. The values and positions of the maxima of cross sections for both processes strongly depend on the electron affinity of an alkaline-earth atom and on the orbital angular momentum l of a highly excited atom. It is shown that in the case of Rydberg atoms in states with large l ～ n ? 1, the rate constants of ion-pair formation and collisional quenching are considerably lower than those for nl-levels with l ? n.     

Fragmentation channels of neutral and charged sodium clusters

The Density Functional Formalism (with the local density approximation for exchange and correlation) is applied to jellium-like spherical particles to obtain heats of fragmentation of Na_n; Na⁺_n and Na²⁺_n (n ⩽ 90) following different decay paths. Masses of the fragments are analyzed resulting that the energetically more favorable channels are controlled by the tendency to fragments with a magic number of electrons, mainly 2 or 8. We have also obtained that, provided we consider parent clusters with the same number of electrons and decay modes involving a neutral fragment, the fragmentations of neutral and ionized clusters are very similar.     

Hybridization and electron interaction in nickel compounds

The bare Coulomb interaction, defined by the reaction 2dⁿ = d^n?1 + d^{n + 1} ? U is determined for some Ni compounds. It is found that U is larger than the 3d bandwidth even for metallic compounds. More relevant to the metal nonmetal transition is the optical gap energy E_G = U ? (E(3d^n?1) ? E(L)) where E(3d^n?1) and E(L) are the d-electron and ligand electron ionization potentials respectively.