Table of contents

We provide, with a new formulation of vector, coherent states for nonlinear spin–orbit Hamiltonian models in terms of the matrix eigenvalue problem for generalized annihilation operators. Nonlinear quaternion vector coherent states are also discussed.

A 'Galois quantum system' in which the position and momentum take values in the Galois field GF(p^ℓ) is considered. It is comprised of ℓ-component systems which are coupled in a particular way and is described by a certain class of Hamiltonians. Displacements in the GF(p^ℓ) × GF(p^ℓ) phase space and the corresponding Heisenberg–Weyl group are studied. Symplectic transformations are shown to form the Sp(2, GF(p^ℓ)) group. Wigner and Weyl functions are defined and their properties are studied. Frobenius symmetries, which are based on Frobenius automorphisms in the theory of Galois fields, are a unique feature of these systems (for ℓ ⩾ 2). If they commute with the Hamiltonian, there are constants of motion which are discussed. An analytic representation in the ℓ-sheeted complex plane provides an elegant formalism that embodies the properties of Frobenius transformations. The difference between a Galois quantum system and other finite quantum systems where the position and momentum take values in the ring is discussed.

Non-equilibrium critical dynamics of the two-dimensional XY model is investigated with Hamiltonian equations of motion. Critical relaxation starting from both ordered and random states is carefully analyzed, and the short-time dynamic scaling behavior is revealed. Logarithmic corrections to scaling are detected for relaxation with a random initial state, while power-law corrections to scaling are observed for relaxation with an ordered initial state. The static exponent η and dynamic exponent z are determined around and below the Kosterlitz–Thouless phase transition temperature. Our results show that the deterministic dynamics described by Hamiltonian equations is in the same universality class as the stochastic dynamics described by Monte Carlo algorithms and Langevin equations.

We study the Casimir effect for a 3D system of ideal Bose gas in a slab geometry with a Dirichlet boundary condition. We calculate the temperature (T) dependence of the Casimir force below and above the Bose–Einstein condensation temperature (T_c). At T ⩽ T_c the Casimir force vanishes as . For T ≳ T_c it weakly depends on temperature. For T ≫ T_c it vanishes exponentially. At finite temperatures this force for thermalized photons in between two plates has a classical expression which is independent of ℏ. At finite temperatures the Casimir force for our system depends on ℏ.

We consider the detection of biased information sources in the ubiquitous code-division multiple-access (CDMA) scheme. We propose a simple modification to both the popular single-user matched-filter detector and a recently introduced near-optimal message-passing-based multiuser detector. This modification allows for detecting modulated biased sources directly with no need for source coding. Analytical results and simulations with excellent agreement are provided, demonstrating substantial improvement in bit error rate in comparison with the unmodified detectors and the alternative of source compression. The robustness of error-performance improvement is shown under practical model settings, including bias estimation mismatch and finite-length spreading codes.

For an ideal D-dimensional Fermi gas under generic external confinement we derive the correcting coefficient (D − 2)/3D of the von Weizsacker term in the kinetic energy density. To obtain this coefficient we use the Kirzhnits semiclassical expansion of the number operator up to the second order in the Planck constant ℏ. Within this simple and direct approach, we determine the differential equation of the density profile and the density functional of the Fermi gas. In the case D = 2, we find that the Kirzhnits gradient corrections vanish to all order in ℏ.

In this paper, we study the mathematical structures of the linear response relation based on Plefka's expansion and the cluster variation method in terms of the perturbation expansion, and we show how this linear response relation approximates the correlation functions of the specified system. Moreover, by comparing the perturbation expansions of the correlation functions estimated by the linear response relation based on these approximation methods with exact perturbative forms of the correlation functions, we are able to explain why the approximate techniques using the linear response relation work well.

The spectral theory of quantum graphs is related via an exact trace formula to the spectrum of the lengths of periodic orbits (cycles) on the graphs. The latter is a degenerate spectrum, and understanding its structure (i.e., finding how many different lengths exist for periodic orbits with a given period and the average number of periodic orbits with the same length) is necessary for the systematic study of spectral fluctuations using the trace formula. This is a combinatorial problem which we solve exactly for complete (fully connected) graphs with arbitrary number of vertices.

We note that the recently introduced fuzzy torus can be regarded as a q-deformed parafermion. Based on this picture, classification of the Hermitian representations of the fuzzy torus is carried out. The result involves Fock-type representations and new finite-dimensional representations for q being a root of unity as well as already known finite-dimensional ones.

A geometrical description of the Lagrangian dynamics in quasi-coordinates on the tangent bundle, using the Lie algebroid framework, is given. Linear non-holonomic systems on Lie algebroids are solved in local coordinates adapted to the constraints, through Lagrangian multipliers and Gibbs–Appell generalized methods.

A complete description of Q-conditional symmetries for two classes of reaction–diffusion–convection equations with power diffusivities is derived. It is shown that all the known results for reaction–diffusion equations with power diffusivities follow as particular cases from those obtained here but not vice versa. The symmetries obtained for constructing exact solutions of the relevant equations are successfully applied. In the particular case, new exact solutions of nonlinear reaction–diffusion–convection equations arising in application and their natural generalizations are found.

We consider the problem of constructing an action functional for physical systems whose classical equations of motion cannot be directly identified with Euler–Lagrange equations for an action principle. Two ways of constructing the action principle are presented. From simple consideration, we derive the necessary and sufficient conditions for the existence of a multiplier matrix which can endow a prescribed set of second-order differential equations with the structure of the Euler–Lagrange equations. An explicit form of the action is constructed if such a multiplier exists. If a given set of differential equations cannot be derived from an action principle, one can reformulate such a set in an equivalent first-order form which can always be treated as the Euler–Lagrange equations of a certain action. We construct such an action explicitly. There exists an ambiguity (not reduced to a total time derivative) in associating a Lagrange function with a given set of equations. We present a complete description of this ambiguity. The general procedure is illustrated by several examples.

The local invariants of a mixed two-qubit system are discussed. These invariants are polynomials in the elements of the corresponding density matrix. They are counted by means of group-theoretic branching rules which relate this problem to one arising in spin–isospin nuclear shell models. The corresponding Molien series and a refinement in the form of a four-parameter generating function are determined. A graphical approach is then used to construct explicitly a fundamental set of 21 invariants. Relations between them are found in the form of syzygies. By using these, the structure of the ring of local invariants is determined, and complete sets of primary and secondary invariants are identified: there are 10 of the former and 15 of the latter.

Let (X, ρ) be a discrete metric space. We suppose that the group acts freely on X and that the number of orbits of X with respect to this action is finite. Then we call X a -periodic discrete metric space. We examine the Fredholm property and essential spectra of band-dominated operators on l^p(X) when 1 < p < ∞. Our approach is based on the theory of band-dominated operators on and their limit operators. In the case where X is the set of vertices of a combinatorial graph, the graph structure defines a Schrödinger operator on l^p(X) in a natural way. We illustrate our approach by determining the essential spectra of Schrödinger operators with slowly oscillating potential both on zig-zag and on hexagonal graphs, the latter being related to nano-structures.

We study finite energy static solutions to a global symmetry breaking Goldstone model described by an isovector scalar field in D + 1 spacetime dimensions. Both topologically stable multisolitons with arbitrary winding numbers and zero topological charge soliton–antisoliton solutions are constructed numerically in D = 3, 4, 5. We have explored the types of symmetries the systems should be subjected to, for there to exist multisoliton and soliton–antisoliton pairs in D = 3, 4, 5, 6. These findings are underpinned by constructing numerical solutions in the D ⩽ 5 examples. Subject to axial symmetry, only multisolitons of all topological charges exist in even D, and in odd D only zero and unit topological charge solutions exist. Subjecting the system to weaker than axial symmetries results in the existence of all the possibilities in all dimensions. Our findings also apply to finite 'energy' solutions to Yang–Mills and Yang–Mills–Higgs systems as well as to sigma models, but we find the numerical work for the Goldstone models more accessible.

Consider in , the operator family . H₀ = a*₁a₁ + ⋅ ⋅ ⋅ + a*_da_d + d/2 is the quantum harmonic oscillator with rational frequencies, W is a symmetric bounded potential, and g is a real coupling constant. We show that if |g| < ρ, ρ being an explicitly determined constant, the spectrum of H(g) is real and discrete. Moreover we show that the operator has a real discrete spectrum but is not diagonalizable.

We investigate the ground state solutions of the Schrödinger equation for complex (non-Hermitian) Hamiltonian systems in two dimensions within the framework of an extended complex phase-space approach. The eigenvalues and eigenfunctions of some two-dimensional complex potentials are found.

We address the problem of unambiguous discrimination among oracle operators. The general theory of unambiguous discrimination among unitary operators is extended with this application in mind. We prove that entanglement with an ancilla cannot assist any discrimination strategy for commuting unitary operators. We also obtain a simple, practical test for the unambiguous distinguishability of an arbitrary set of unitary operators on a given system. Using this result, we prove that the unambiguous distinguishability criterion is the same for both standard and minimal oracle operators. We then show that, except in certain trivial cases, unambiguous discrimination among all standard oracle operators corresponding to integer functions with fixed domain and range is impossible. However, we find that it is possible to unambiguously discriminate among the Grover oracle operators corresponding to an arbitrarily large unsorted database. The unambiguous distinguishability of standard oracle operators corresponding to totally indistinguishable functions, which possess a strong form of classical indistinguishability, is analysed. We prove that these operators are not unambiguously distinguishable for any finite set of totally indistinguishable functions on a Boolean domain and with arbitrary fixed range. Sets of such functions on a larger domain can have unambiguously distinguishable standard oracle operators, and we provide a complete analysis of the simplest case, that of four functions. We also examine the possibility of unambiguous oracle operator discrimination with multiple parallel calls and investigate an intriguing unitary superoperator transformation between standard and entanglement-assisted minimal oracle operators.

In this work, we study the two-point entanglement S(i, j), which measures the entanglement between two separated degrees of freedom (ij) and the rest of system, near a quantum phase transition. Away from the critical point, S(i, j) saturates with a characteristic length scale ξ_E, as the distance |i − j| increases. The entanglement length ξ_E agrees with the correlation length. The universality and finite size scaling of entanglement are demonstrated in a class of exactly solvable one-dimensional spin model. By connecting the two-point entanglement to correlation functions in the long range limit, we argue that the prediction power of a two-point entanglement is universal as long as the two involved points are separated far enough.

The existence of a family of coherent states (CS) solving the identity in a Hilbert space allows, under certain conditions, to quantize functions defined on the measure space of CS parameters. The application of this procedure to the 2-sphere provides a family of inequivalent CS quantizations based on the spin spherical harmonics (the CS quantization from usual spherical harmonics appears to give a trivial issue for the Cartesian coordinates). We compare these CS quantizations to the usual (Madore) construction of the fuzzy sphere. Due to these differences, our procedure yields new types of fuzzy spheres. Moreover, the general applicability of CS quantization suggests similar constructions of fuzzy versions of a large variety of sets.

In this paper we give a method to associate a graph with an arbitrary density matrix referred to a standard orthonormal basis in the Hilbert space of a finite dimensional quantum system. We study related issues such as classification of pure and mixed states, Von Neumann entropy, separability of multipartite quantum states and quantum operations in terms of the graphs associated with quantum states. In order to address the separability and entanglement questions using graphs, we introduce a modified tensor product of weighted graphs, and establish its algebraic properties. In particular, we show that Werner's definition (Werner 1989 Phys. Rev. A 40 4277) of a separable state can be written in terms of graphs, for the states in a real or complex Hilbert space. We generalize the separability criterion (degree criterion) due to Braunstein et al (2006 Phys. Rev. A 73 012320) to a class of weighted graphs with real weights. We have given some criteria for the Laplacian associated with a weighted graph to be positive semidefinite.

We outline a remarkably efficient method for generating solutions to quantum anharmonic oscillators with an x^2M potential. We solve the Schrödinger equation in terms of a free parameter which is then tuned to give the correct boundary condition by generating a power series expansion of the wavefunction in x and applying a modified Borel resummation technique to obtain the large x behaviour. The process allows us to calculate energy eigenvalues to an arbitrary level of accuracy. High degrees of precision are achieved even with modest computing power. Our technique extends to all levels of excitation and produces the correct solution to the double well oscillators even though they are dominated by non-perturbative effects.

We prove that the nonorthogonal states randomly selected from a set can evolve into a linear superposition of multiple original and orthogonal complementing states with failure branch if and only if the input states are linearly independent. The results for a single-input state are also generalized into the case of several copies of an input state.

Using the technique of integration within the ordered product (IWOP) of operators, we show that the operator U = exp [ir(∑ⁿ⁻¹_i=1Q_iP_i+1 + Q_nP₁)] is an N-mode squeezing operator for the N-mode quadratures exhibiting the standard squeezing. The corresponding squeezed vacuum state in N-mode Fock space is derived, and the entanglement involved in such a state is also explained. We present an optical network for producing the N-mode squeezed state.

The spectra of a particular class of -symmetric eigenvalue problems has previously been studied, and found to have an extremely rich structure. In this paper, we present an explanation for these spectral properties in terms of quantization conditions obtained from the complex WKB method. In particular, we consider the relation of the quantization conditions to the reality and positivity properties of the eigenvalues. The methods are also used to examine further the pattern of eigenvalue degeneracies observed by Dorey et al (2001 J. Phys. A: Math. Gen.34 5679 (Preprint hep-th/0103051), 2001 J. Phys. A: Math. Gen.34 L391 (Preprint hep-th/0104119)).

We study a system of electrons moving on a noncommutative plane in the presence of an external magnetic field which is perpendicular to this plane. For generality we assume that the coordinates and the momenta are both noncommutative. We make a transformation from the noncommutative coordinates to a set of commuting coordinates and then we write the Hamiltonian for this system. The energy spectrum and the expectation value of the current can then be calculated and the Hall conductivity can be extracted. We use the same method to calculate the phase shift for the Aharonov–Bohm effect. Precession measurements could allow strong upper limits to be imposed on the noncommutativity coordinate and momentum parameters Θ and Ξ.

We investigate the properties of chiral anomalies in d = 2 in the framework of constructive quantum field theory. The condition that the gauge propagator is sufficiently soft in the ultraviolet is essential for the anomaly non-renormalization; when it is violated, as for contact current–current interactions, the anomaly is renormalized by higher order corrections. The same conditions are also essential for the validity, in the massless case of the closed equation obtained combining Ward identities and Schwinger–Dyson equations; this solves the apparent contradiction between perturbative computations and exact analysis.

Within the spirit of Dirac's canonical quantization, noncommutative spacetime field theories are introduced by making use of the reparametrization invariance of the action and of an arbitrary non-canonical symplectic structure. This construction implies that the constraints need to be deformed, resulting in an automatic Drinfeld twisting of the generators of the symmetries associated with the reparametrized theory. We illustrate our procedure for the case of a scalar field in (1+1)-spacetime dimensions, but it can be readily generalized to arbitrary dimensions and arbitrary types of fields.

Equilibrium configurations of dusty plasmas with grains of different sizes, which interact through a screened Coulomb force field and confined by a two-dimensional quadratic potential, are studied using molecular dynamics simulation. The system configuration depends on the sizes, masses and charges of the grain species as well as the screening strength of the background plasma. The consideration of the grain size has established a different equilibrium configuration relative to that of point grains. In the new configurations, grains of different species separate into different shells, with the grains of larger mass and charge located away from the system center, forming a shell that surrounds the grains of smaller mass and charge at the system center. This configuration occurs beyond a critical grain radius, and its structure and size are determined by the competing effects between the inter-grain electrostatic repulsive force, the screening effect of the plasma and the mass-dependent confinement force of the quadratic potential.

The lattice Boltzmann model for the nonlinear Schrödinger equation is proposed. The new model is based on the technique of the higher order moment of equilibrium distribution functions and a series of lattice Boltzmann equations in different time scales. The Euler equations are derived from the nonlinear Schrödinger equation by removing non-physical pressure. We have simulated two irrotational flows. These numerical results agree well with classical ones.