Correlations in a General Theory of Quantum Measurement

The paper investigates correlations in a general theory of quantum measurement based on the notion of instrument. The analysis is performed in the algebraic formalism of quantum theory in which the observables of a physical system are described by a von Neumann algebra, and the states—by normal positive normalized functionals on this algebra. The results extend and generalise those obtained for the classical case where one deals with the full algebra of operators on a Hilbert space.     

An algebraic method for Schrödinger equations in quaternionic quantum mechanics

In the study of theory and numerical computations of quaternionic quantum mechanics and quantum chemistry, one of the most important tasks is to solve the Schrödinger equation with A an anti-self-adjoint real quaternion matrix, and |f〉 an eigenstate to A. The quaternionic Schrödinger equation plays an important role in quaternionic quantum mechanics, and it is known that the study of the quaternionic Schrödinger equation is reduced to the study of quaternionic eigen-equation Aα=αλ with A an anti-self-adjoint real quaternion matrix (time-independent). This paper, by means of complex representation of quaternion matrices, introduces concepts of norms of quaternion matrices, studies the problems of quaternionic Least Squares eigenproblem, and give a practical algebraic technique of computing approximate eigenvalues and eigenvectors of a quaternion matrix in quaternionic quantum mechanics.     

Stochastic Schrödinger Equations as Limit of Discrete Filtering

We consider an open model possessing a Markovian quantum stochastic limit and derive the limit stochastic Schrödinger equations for the wave function conditioned on indirect observations using only the von Neumann projection postulate. We show that the diffusion (Gaussian) situation is universal as a result of the central limit theorem with the quantum jump (Poissonian) situation being an exceptional case. It is shown that, starting from the correponding limiting open systems dynamics, the theory of quantum filtering leads to the same equations, therefore establishing consistency of the quantum stochastic approach for limiting Markovian models.     

On Ideal Measurements and Their Corresponding Instruments on von Neumann Algebras

We study some features of quantum measurement in the framework of the theory of instruments — a mathematical model for measurement theory. This investigation is carried in an algebraic formalism in which the observables are represented by elements of a von Neumann algebra and the states are linear normal positive functionals on this algebra. We give a characterization of ideal instruments and study their connections with weakly and strongly repeatable instruments under the assumption that the associated observables are projection-valued measures. We also show that the observables of ideal weakly repeatable instruments must be projection-valued measures.     

Quantum Set Theory Extending the Standard Probabilistic Interpretation of Quantum Theory

The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality between two arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the standard probabilistic interpretation of quantum theory to define the probability of equality between two arbitrary observables in an arbitrary state. We apply this new interpretation to quantum measurement theory, and establish a logical basis for the difference between simultaneous measurability and simultaneous determinateness.     

Liouville Invariance in Quantum and Classical Mechanics

The density-matrix and Heisenberg formulations of quantum mechanics follow—for unitary evolution—directly from the Schrödinger equation. Nevertheless, the symmetries of the corresponding evolution operator, the Liouvillian L = i[, H], need not be limited to those of the Hamiltonian H. This is due to L only involving eigenenergy differences, which can be degenerate even if the energies themselves are not. Remarkably, this possibility has rarely been mentioned in the literature, and never pursued more generally. We consider an example involving mesoscopic Josephson devices, but the analysis only assumes familiarity with basic quantum mechanics. Subsequently, such L-symmetries are shown to occur more widely, in particular also in classical mechanics. The symmetry's relevance to dissipative systems and quantum-information processing is briefly discussed. PACS: 03.65.-w, 03.67.-a, 45.20.Jj, 74.50.+r     

Parallel Split-Step Fourier Methods for the Coupled Nonlinear Schrödinger Type Equations

The nonlinear Schrödinger type equations are of tremendous interest in both theory and applications. Various regimes of pulse propagation in optical fibers are modeled by some form of the nonlinear Schrödinger equation.In this paper we introduce parallel split-step Fourier methods for the numerical simulations of the coupled nonlinear Schrödinger equation that describes the propagation of two orthogonally polarized pulses in a monomode birefringent fibers. These methods are implemented on the Origin 2000 multiprocessor computer. Our numerical experiments have shown that these methods give accurate results and considerable speedup.     

How to Steer a Quantum System over a Schrödinger Bridge

A new approach to the steering problem for the Schrödinger equation relying on stochastic mechanics and on the theory of Schrödinger bridges is presented. Given the initial and final states ₀ and ₁, respectively, the desired quantum evolution is constructed with the aid of a reference quantum evolution. The Nelson process corresponding to the latter evolution is used as reference process in a Schrödinger bridge problem with marginal probability densities | ₀|² and | ₁|². This approach is illustrated by working out a simple Gaussian example. PACS: 03.65.-w     

Three-dimensional quantum simulation of multigate nanowire field effect transistors

Detailed numerical methods for the three-dimensional quantum simulation of the multigate nanowire field effect transistors in the ballistic transport regime are presented in this work. The device has been modeled based on the effective mass theory and the non-equilibrium Green’s function formalism, and its simulation consists of solutions of the three-dimensional Poisson’s equation, two-dimensional Schrödinger equations on the cross-sectional planes, and one-dimensional transport equation. Details on numerical techniques for each of the simulation steps are described, with a special attention to the solution of the most CPU demanding two-dimensional Schrödinger equation.     

Nonlinear stability,excitation and soliton solutions in electrified dispersive systems

A weakly nonlinear theory of wave propagation in two superposed dielectric fluids in the presence of a horizontal electric field is investigated in (2+1)-dimensions. The equation governing the evolution of the amplitude of the progressive waves is obtained in the form of a two-dimensional nonlinear Schrödinger equation. A three-wave resonant interaction for nonlinear excitations created from electrohydrodynamic capillary-gravity waves is observed to be possible in a dispersive medium with a self-focusing cubic nonlinearity. Under suitable conditions, the nonlinear envelope equations for the resonant interaction are derived by using multiple scales and inverse scattering methods, and an explicit three-wave soliton solution is discussed. Both the dynamic properties and the modulational instability of finite amplitude electrohydrodynamic wave are studied for the cubic nonlinear Schrödinger equation by means of linearized stability analysis and the nonlinear interaction coefficient. We show that the trajectories in phase space exhibit different behavior with the increase of nonlinear perturbations, and we determine the electric field and wavenumber ranges at which the original point is elliptic or hyperbolic, respectively. It is found also that the presence of the electric field in the equation modifies the nature of wave stability and soliton structures, and that the amplitude and width of the soliton are decreased and increased, respectively, when the electric field value increases.     

Quantum covariance and filtering

We give a tutorial exposition of the analogue of the filtering equation for quantum systems focusing on the quantum probabilistic framework and developing the ideas from the classical theory. Quantum covariances and conditional expectations on von Neumann algebras play an essential part in the presentation.     

Linear and Nonlinear Dynamical Chaos

Interrelations between dynamical and statistical laws in physics on the one hand, and between the classical and quantum mechanics on the other hand, are discussed with the emphasis on the new phenomenon of dynamical chaos.The principal results of the studies into chaos in classical mechanics are presented in some detail within the general picture of chaos as a specific case of dynamical behavior. These results include the strong local instability and robustness of motion, continuity of both the phase space as well as the motion spectrum, and time reversibility but nonrecurrency of statistical evolution.The analysis of apparently very deep and challenging contradictions of this picture with the quantum principles is given. The quantum view of dynamical chaos, as an attempt to resolve these contradictions guided by the correspondence principle and based upon the characteristic time scales of quantum evolution, is explained. The picture of the quantum chaos as a new generic dynamical phenomenon is outlined together with a few other examples of such a chaos, including linear (classical) waves and a digital computer.I conclude with the discussion of two fundamental physical problems: the quantum measurement (-collapse) and the causality principle, which both appear to be related to the phenomenon of dynamical chaos.     

Solution of the Schrödinger equation over an infinite integration interval by perturbation methods, revisited

We consider the solution of the one-dimensional Schrödinger problem over an infinite integration interval. The infinite problem is regularized by truncating the integration interval and imposing the appropriate boundary conditions at the truncation points. The Schrödinger problem is then solved on the truncated integration interval using one of the piecewise perturbation methods developed for the regular Schrödinger problem.We select the truncation points using a procedure based on the WKB approximation. However for problems which behave as a Coulomb problem both around the origin and in the asymptotic range, a more accurate treatment of the numerical boundaries is possible. Taking into account the asymptotic form of the Coulomb equation, adapted boundary conditions can be constructed and as a consequence smaller truncation points can be chosen. To deal with the singularity of the Coulomb-like problem around the origin, a special perturbative algorithm is applied in a small interval around the origin.     

Non-perturbative time-dependent numerical simulation for electromagnetically induced transparency and slow light in three-level Λ-type atoms

A time-dependent, semi-classical simulation model of light coupling with an array of three-level Λ-type atoms is developed to study dynamical phenomena of electromagnetically induced transparency (EIT). Light is described by electromagnetic wave equation and the quantum dynamics of atoms is described by time-dependent non-perturbative Schrödinger equation. Dynamical pictures and the mechanism of EIT are demonstrated and this model is used to explore the light–matter interaction of ultra-short pulse.     

Application of coordinate transformation and finite differences method in numerical modeling of quantum dash band structure

In this paper we propose an efficient and simple method for the band structure calculation of semiconductor quantum dashes. The method combines a coordinate transformation (mapping) based on an analytical function and the finite differences method (FDM) for solving the single-band Schrödinger equation. We explore suitable coordinate transformations and propose those, which might simultaneously provide a satisfactory fit of the quantum dash heterointerface and creation of an appropriate computational domain which encloses the quantum dash structure. After mapping of the quantum dash and the rest of computational domain, the Schrödinger equation is solved by the FDM in the mapped space. For the proposed coordinate transformations, we investigate and analyze applicability, robustness and convergence of the method by varying the FDM grid density and size of the computational domain. We find that the method provides sufficient accuracy, stability and flexibility with respect to the size and shape of the quantum dash and above all, extreme simplicity, which is promising and essential for an extension of the method to the multiband Schrödinger equation case.     

Impact of large cutoff-effects on algorithms for improved Wilson fermions

As a feasibility study for a scaling test we investigate the behavior of algorithms for dynamical fermions in the Nf=2 Schrödinger functional at an intermediate volume of 1 fm⁴. Simulations were performed using HMC with two pseudo-fermions and PHMC at lattice spacings of approximately 0.1 and 0.07 fm. We show that some algorithmic problems are due to large cutoff-effects in the spectrum of the improved Wilson-Dirac operator and disappear at the smaller lattice spacing. The problems discussed here are not expected to be specific to the Schrödinger functional.     

Solving Sturm-Liouville problems by piecewise perturbation methods, revisited

We present the extension of the successful Constant Perturbation Method (CPM) for Schrödinger problems to the more general class of Sturm-Liouville eigenvalue problems. Whereas the original CPM can only be applied to Sturm-Liouville problems after a Liouville transformation, the more general CPM presented here solves the Sturm-Liouville problem directly. This enlarges the range of applicability of the CPM to a wider variety of problems and allows a more efficient solution of many problems. The CPMs are closely related to the second-order coefficient approximation method underlying the SLEDGE software package, but provide for higher order approximations. These higher order approximations can also be obtained by applying a modified Neumann method. The CPM approach, however, leads to simpler formulae in a more convenient form.     

Kernel-based classification using quantum mechanics

This paper introduces a new nonparametric estimation approach inspired from quantum mechanics. Kernel density estimation associates a function to each data sample. In classical kernel estimation theory the probability density function is calculated by summing up all the kernels. The proposed approach assumes that each data sample is associated with a quantum physics particle that has a radial activation field around it. Schrödinger differential equation is used in quantum mechanics to define locations of particles given their observed energy level. In our approach, we consider the known location of each data sample and we model their corresponding probability density function using the analogy with the quantum potential function. The kernel scale is estimated from distributions of K-nearest neighbours statistics. In order to apply the proposed algorithm to pattern classification we use the local Hessian for detecting the modes in the quantum potential hypersurface. Each mode is assimilated with a nonparametric class which is defined by means of a region growing algorithm. We apply the proposed algorithm on artificial data and for the topography segmentation from radar images of terrain.