Table of contents

We give a formal algebraic description of Josephson-type quantum dynamical systems, i.e., Hamiltonian systems with a cos ϑ-like potential term. The two-boson Heisenberg algebra plays for such systems the role that the h(1) algebra does for the harmonic oscillator. A single Josephson junction is selected as a representative of Josephson systems. We construct both logical states (codewords) and logical (gate) operators in the superconductive regime. The codewords are the even and odd coherent states of the two-boson algebra: they are shift-resistant and robust, due to squeezing. The logical operators acting on the qubit codewords are expressed in terms of operators in the enveloping of the two-boson algebra. Such a scheme appears to be relevant for quantum information applications.

We show that the convex set of separable mixed states of the 2 × 2 system is a body of a constant height. This fact is used to prove that the probability of finding a random state to be separable equals twice the probability of finding a random boundary state to be separable, provided that the random states are generated uniformly with respect to the Hilbert–Schmidt (Euclidean) measure. An analogous property holds for the set of positive-partial-transpose states for an arbitrary bipartite system.

We present in this paper an exact study of a first-order transition induced by an inhomogeneous boundary magnetic field in the 2D Ising model. From a previous analysis of the interfacial free energy in the discrete case (Clusel and Fortin 2005 J. Phys. A: Math. Gen.38 2849) we identify, using an asymptotic expansion in the thermodynamic limit, the line of transition that separates the regime where the interface is localized near the boundary from the one where it is propagating inside the bulk. In particular, the transition line has a strong dependence on the aspect ratio of the lattice.

We develop techniques to obtain rigorous bounds on the behaviour of random walks on combs. Using these bounds, we calculate exactly the spectral dimension of random combs with infinite teeth at random positions or teeth with random but finite length. We also calculate exactly the spectral dimension of some fixed non-translationally invariant combs. We relate the spectral dimension to the critical exponent of the mass of the two-point function for random walks on random combs, and compute mean displacements as a function of walk duration. We prove that the mean first passage time is generally infinite for combs with anomalous spectral dimension.

A new method of diagonalization of the XY-Hamiltonian of inhomogeneous open linear chains with periodic (in space) changing Larmor frequencies and coupling constants is developed. As an example of application, multiple-quantum dynamics of an inhomogeneous chain consisting of 1001 spins is investigated. Intensities of multiple-quantum coherences are calculated for arbitrary inhomogeneous chains in the approximation of the next-nearest interactions.

We study self-adjoint operators defined by factorizing second-order differential operators in first-order ones. We discuss examples where such factorizations introduce singular interactions into simple quantum-mechanical models such as the harmonic oscillator or the free particle on the circle. The generalization of these examples to the many-body case yields quantum models of distinguishable and interacting particles in one dimensions which can be solved explicitly and by simple means. Our considerations lead us to a simple method to construct exactly solvable quantum many-body systems of Calogero–Sutherland type.

We extend the exact periodic Bethe ansatz solution for one-dimensional bosons and fermions with δ-interaction and arbitrary internal degrees of freedom to the case of hard wall boundary conditions. We give an analysis of the ground-state properties of fermionic systems with two internal degrees of freedom, including expansions of the ground-state energy in the weak and strong coupling limits.

A deformation parameter of a bihamiltonian structure of hydrodynamic type is shown to parametrize different extensions of the AKNS hierarchy to include negative flows. This construction establishes a purely algebraic link between, on the one hand, two realizations of the first negative flow of the AKNS model and, on the other, two-component generalizations of Camassa–Holm- and Dym-type equations. The two-component generalizations of Camassa–Holm- and Dym-type equations can be obtained from the negative-order Hamiltonians constructed from the Lenard relations recursively applied on the Casimir of the first Poisson bracket of hydrodynamic type. The positive-order Hamiltonians, which follow from the Lenard scheme applied on the Casimir of the second Poisson bracket of hydrodynamic type, are shown to coincide with the Hamiltonians of the AKNS model. The AKNS Hamiltonians give rise to charges conserved with respect to equations of motion of two-component Camassa–Holm- and two-component Dym-type equations.

We apply a recently introduced reduction procedure based on the embedding of non-crystallographic Coxeter groups into crystallographic ones to Calogero–Moser systems. For rational potentials the familiar generalized Calogero Hamiltonian is recovered. For the Hamiltonians of trigonometric, hyperbolic and elliptic types, we obtain novel integrable dynamical systems with a second potential term which is rescaled by the golden ratio. We explicitly show for the simplest of these non-crystallographic models, how the corresponding classical equations of motion can be derived from a Lie algebraic Lax pair based on the larger, crystallographic Coxeter group.

We show that the dynamics of a birational map on an elliptic curve over a field is, typically, conjugate to addition by a point (under the associated group law). When the field is taken to be the function field of rational complex functions of one variable, this amounts to an algebraic geometric version of the Arnold–Liouville integrability theorem for planar integrable maps. By-products of this approach are that birational maps preserving foliations are necessarily the composition of two involutions, and that relationships between birational maps preserving the same foliation can be described in terms of the respective points they add on the corresponding Weierstrass curves. When the result is applied to finite fields, it helps explain some universal features of the periodic orbit distribution function for the reductions of integrable maps.

We show that the third-order difference equations proposed by Hirota, Kimura and Yahagi are generated by a pair of second-order difference equations. In some cases, the pair of the second-order equations are equivalent to the Quispel–Robert–Thomson (QRT) system, but in the other cases, they are irrelevant to the QRT system. We also discuss an ultradiscretization of the equations.

The performance of the positive P phase-space representation for exact many-body quantum dynamics is investigated. Gases of interacting bosons are considered, where the full quantum equations to simulate are of a Gross–Pitaevskii form with added Gaussian noise. This method gives tractable simulations of many-body systems because the number of variables scales linearly with the spatial lattice size. An expression for the useful simulation time is obtained, and checked in numerical simulations. The dynamics of first-, second- and third-order spatial correlations are calculated for a uniform interacting 1D Bose gas subjected to a change in scattering length. Propagation of correlations is seen. A comparison is made with other recent methods. The positive P method is particularly well suited to open systems as no conservation laws are hard-wired into the calculation. It also differs from most other recent approaches in that there is no truncation of any kind.

An algebraic approach to class of separable non-central Hamiltonians is presented. We show that the bound states of quantum systems under consideration are described by unitary representations of the algebra.

A countable set of asymptotic space-localized solutions is constructed by the complex germ method in the adiabatic approximation for the nonstationary Gross–Pitaevskii equation with nonlocal nonlinearity and a quadratic potential. The asymptotic parameter is 1/T, where T ≫ 1 is the adiabatic evolution time. A generalization of the Berry phase of the linear Schrödinger equation is formulated for the Gross–Pitaevskii equation. For the solutions constructed, the Berry phases are found in explicit form.

We investigate the motion of bound-state poles in two quantum wave guides laterally coupled through a window. The imaginary momentum ik at the bound-state poles is studied as a function of the size a of the window. Both bound and virtual states appear as a spans the whole range from 0 up to +∞. We are able to find simple scaling laws relating the critical value of the window size at which the nth bound state appears to the number n of bound states, in the limit of large n. A similar relation is also provided for the slope and the second derivative of the pole trajectories in the (k, a) plane. These relations are characterized by an extremely high numerical accuracy. We also evaluate the exact value of the first two derivatives for a = 0.

The authors would like to correct equations (81a) and (81b). Please see PDF for corrected equations.