Quantum statistics of the degenerate parametric amplifier

From the quantum mechanics and the quantum optics point of view the problem of the time-dependent Hamiltonian which describes the degenerate parametric amplifier is presented. Under a certain integrability condition the solution of the Heisenberg equations of motion is given. The wave function for both the Schrödinger picture and quasi-coherent states, as well as the Green's function is obtained. The Glauber second order correlation function, and the squeezing and higher order squeezing is examined. The quasi-probability distribution function is also considered.     

Photon statistics and self-induced gyrotropic birefringence

The influence of the self-induced optical anisotropy appearing in the wings of a degenerate dipole transition on the quantum properties of elliptically or linearly polarized monochromatic light is calculated in the Heisenberg picture. Further the photon statistics (bunching or antibunching) of the weak field detected after passing a phase retarder and a polarizer is studied. Finally the possibility to measure small deviations from the Poisson distribution without coincidence experiments is discussed.     

Deformation quantization of Heisenberg manifolds

ForM a smooth manifold equipped with a Poisson bracket, we formulate aC*-algebra framework for deformation quantization, including the possibility of invariance under a Lie group of diffeomorphisms preserving the Poisson bracket. We then show that the much-studied non-commutative tori give examples of such deformation quantizations, invariant under the usual action of ordinary tori. Going beyond this, the main results of the paper provide a construction of invariant deformation quantizations for those Poisson brackets on Heisenberg manifolds which are invariant under the action of the Heisenberg Lie group, and for various generalizations suggested by this class of examples. Interesting examples are obtained of simpleC*-algebras on which the Heisenberg group acts ergodically.This work was supported in part by National Science Foundation grant DMS 8601900     

Fractal and lacunary stochastic processes

Discrete-time random walks simulate diffusion if the single-step probability density function (jump distribution) generating the walk is sufficiently shortranged. In contrast, walks with long-ranged jump distributions considered in this paper simulate Lévy or stable processes. A one-dimensional walk with a selfsimilar jump distribution (the Weierstrass random walk) and its higherdimensional generalizations generate fractal trajectories if certain transience criteria are met and lead to simple analogs of deep results on the Hausdorff-Besicovitch dimension of stable processes. The Weierstrass random walk is lacunary (has gaps in the set of allowed steps) and its characteristic function is Weierstrass' non-differentiable function. Other lacunary random walks with characteristic functions related to Riemann's zeta function and certain numbertheoretic functions have very interesting analytic structure.     

Spin-glass transition in bond-disordered Heisenberg antiferromagnets coupled with local lattice distortions on a pyrochlore lattice

Motivated by puzzling characteristics of spin-glass transitions widely observed in pyrochlore-based frustrated materials, we investigate the effects of coupling to local lattice distortions in a bond-disordered antiferromagnet on the pyrochlore lattice by extensive Monte Carlo simulations. We show that the spin-glass transition temperature T(f) is largely enhanced by the spin-lattice coupling and, furthermore, becomes almost independent of Δ in a wide range of the disorder strength Δ. The critical property of the spin-glass transition is indistinguishable from that of the canonical Heisenberg spin glass in the entire range of Δ. These peculiar behaviors are ascribed to a modification of the degenerate manifold from a continuous to semidiscrete one by spin-lattice coupling.     

Spin diffusion in a Heisenberg spin glass

Using a combination of analytic and Monte Carlo techniques we obtain estimates for the spin diffusion constant and spin conductivity in a classical simple cubic Heisenberg spin glass which has a Gaussian distribution of exchange interactions between nearest neighbors.     

Motion of bound domain walls in a spin ladder

The elementary excitation spectrum of the spin- \frac12\frac{1}{2} antiferromagnetic (AFM) Heisenberg chain is described in terms of a pair of freely propagating spinons. In the case of the Ising-like Heisenberg Hamiltonian spinons can be interpreted as domain walls (DWs) separating degenerate ground states. In dimension d > 1, the issue of spinons as elementary excitations is still unsettled. In this paper, we study two spin- \frac12\frac{1}{2} AFM ladder models in which the individual chains are described by the Ising-like Heisenberg Hamiltonian. The rung exchange interactions are assumed to be pure Ising-type in one case and Ising-like Heisenberg in the other. Using the low-energy effective Hamiltonian approach in a perturbative formulation, we show that the spinons are coupled in bound pairs. In the first model, the bound pairs are delocalized due to a four-spin ring exchange term in the effective Hamiltonian. The appropriate dynamic structure factor is calculated and the associated lineshape is found to be almost symmetric in contrast to the 1d case. In the case of the second model, the bound pair of spinons lowers its kinetic energy by propagating between chains. The results obtained are consistent with recent theoretical studies and experimental observations on ladder-like materials.     

Random walk representations of the Heisenberg model

We develop random walk representations for the spin-S Heisenberg ferromagnet with nearest neighbor interactions. We show that the spin-S Heisenberg model is a diffusion with local times controlled by the spin-S Ising model. As a consequence, expectations for the Heisenberg model conditioned on zero diffusion are shown to be Ising expectations.     

Bipartite entanglement in the spin-1/2 Ising-Heisenberg planar lattice constituted of identical trigonal bipyramidal plaquettes

The bipartite entanglement is rigorously examined in the spin-1/2 Ising-Heisenberg planar lattice composed of identical inter-connected bipyramidal plaquettes at zero and finite temperatures using the quantity called concurrence. It is shown that the Heisenberg spins of the same plaquette are twice stronger entangled in the two-fold degenerate quantum ground state than in the macroscopically degenerate quantum chiral one. The bipartite entanglement with chiral features completely disappears below or exactly at the critical temperature of the model, while that with no chirality may survive even above the critical temperature of the model. Non-monotonous temperature variations of the concurrence clearly evidence the activation of the entangled Heisenberg states also above classical ground state as well as their re-appearance above the critical temperature of the model.     

An action function for a higher step Grushin operator

The purpose of this paper is to discuss how we can construct the heat kernel for (sub)-Laplacian in an explicit (integral) form in terms of a certain class of special functions. Of course, such cases will be highly limited. Here we only treat a typical operator, called Grushin operator. So, first we explain two methods to construct the heat kernel of a “step 2” Grushin operator. One is the eigenfunction expansion which leads to an integral form for the heat kernel, then we treat the formula by a method called, complex Hamilton–Jacobi method invented by Beals–Gaveau–Greiner. One of the main result in this paper is to construct an action function for a higher order oscillator. Until now, no explicit expression of the heat kernel for higher order cases have been given in an explicit form and we show a phenomenon that our action function will play a role toward the construction of the heat kernel of higher step Grushin operators.     

Uniaxially anisotropic antiferromagnets in a field on a square lattice

Classical uniaxially anisotropic Heisenberg and XY antiferromagnets in a field along the easy axis on a square lattice are analysed, applying ground state considerations and Monte Carlo techniques. The models are known to display antiferromagnetic and spin-flop phases. In the Heisenberg case, a single-ion anisotropy is added to the XXZ antiferromagnet, enhancing or competing with the uniaxial exchange anisotropy. Its effect on the stability of non-collinear structures of biconical type is studied. In the case of the anisotropic XY antiferromagnet, the transition region between the antiferromagnetic and spin-flop phases is found to be dominated by degenerate bidirectional fluctuations. The phase diagram is observed to resemble closely that of the XXZ antiferromagnet without single-ion anisotropy.     

A statistical framework for the classification of tensor morphologies in diffusion tensor images

Tractography algorithms for diffusion tensor (DT) images consecutively connect directions of maximal diffusion across neighboring DTs in order to reconstruct the 3-dimensional trajectories of white matter tracts in vivo in the human brain. The performance of these algorithms, however, is strongly influenced by the amount of noise in the images and by the presence of degenerate tensors-- i.e., tensors in which the direction of maximal diffusion is poorly defined. We propose a simple procedure for the classification of tensor morphologies that uses test statistics based on invariant measures of DTs, such as fractional anisotropy, while accounting for the effects of noise on tensor estimates. Examining DT images from seven human subjects, we demonstrate that this procedure validly classifies DTs at each voxel into standard types (nondegenerate DTs, as well as degenerate oblate, prolate or isotropic DTs), and we provide preliminary estimates for the prevalence and spatial distribution of degenerate tensors in these brains. We also show that the P values for test statistics are more sensitive tools for classifying tensor morphologies than are invariant measures of anisotropy alone.     

A meshless method based on moving Kriging interpolation for a two-dimensional time-fractional diffusion equation

Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the moving Kriging inter- polation is developed for a two-dimensional time-fractional diffusion equation. The shape function and its derivatives are obtained by the moving Kriging interpolation technique. For possessing the Kronecker delta property, this technique is very efficient in imposing the essential boundary conditions. The governing time-fractional diffusion equations are transformed into a standard weak formulation by the Galerkin method. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard central difference method. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in detail.     

Phase transitions in anisotropic lattice spin systems

A general method for proving the existence of phase transitions is presented and applied to six nearest neighbor models, both classical and quantum mechanical, on the two dimensional square lattice. Included are some two dimensional Heisenberg models. All models are anisotropic in the sense that the groundstate is only finitely degenerate. Using our method which combines a Peierls argument with reflection positivity, i.e. chessboard estimates, and the principle of exponential localization we show that five of them have long range order at sufficiently low temperature. A possible exception is the quantum mechanical, anisotropic Heisenberg ferromagnet for which reflection positivity isnot proved, but for which the rest of the proof is valid.Work partially supported by U.S. National Science Foundation grant no. MPS 75-11864Work partially supported by U.S. National Science Foundation grant no. MCS 75-21684 A01     

A criterion for global hyperbolicity of left-invariant Lorentz metrics on Lie groups

We show that every left-invariant Lorentz metric on a non-abelian simply connected Lie group is globally hyperbolic whenever its restriction to the commutator ideal of the Lie algebra is positive definite. We also show that a left-invariant Lorentz metric on the three-dimensional Heisenberg group is globally hyperbolic if and only if its restriction to the center of the Lie algebra is positive definite or degenerate.     

A Model for the Shapes of Advected Triangles

Three particles floating on a fluid surface define a triangle. The aim of this paper is to characterise the shape of the triangle, defined by two of its angles, as the three vertices are subject to a complex or turbulent motion. We consider a simple class of models for this process, involving a combination of a random strain of the fluid and Brownian motion of the particles. Following D.G. Kendall, we map the space of triangles to a sphere, whose equator corresponds to degenerate triangles with colinear vertices, with equilaterals at the poles. We map our model to a diffusion process on the surface of the sphere and find an exact solution for the shape distribution. Whereas the action of the random strain tends to make the shape of the triangles infinitely elongated, in the presence of a Brownian diffusion of the vertices, the model has an equilibrium distribution of shapes. We determine here exactly this shape distribution in the simple case where the increments of the strain are diffusive.     

Quantum breathers in a finite Heisenberg spin chain with antisymmetric interactions

A study of the likelihood of quantum breathers in a quantum Heisenberg spin system including a Dzyaloshinsky-Moriya interaction (DMI) is done through an extended Bose-Hubbard model while using the scheme of few body physics. The energy spectrum of the resulting Bose-Hubbard Hamiltonian, on a periodic one-dimensional lattice containing more than two quanta shows interesting detailed band structures. From a non degenerate, and a degenerate perturbation theory in addition to a numerical diagonalization, a careful investigation of these fine structures is set up. The attention is focussed on the effects of various interactions that are; the DMI, the Heisenberg in-plane (X, Y) as well as the out of plane exchange interaction on the energy spectrum of such a system. The outcome displays a possibility of an energy self-compensation in the system. We also computed the weight function of the eigenstates in direct space and in the space of normal modes. From a perturbation theory it is shown that the interaction between the quanta leads to an algebraic localization of the modified extended states in the normal-mode space of the non-interacting system that are coined quantum q-breathers excitations.     

Spin Transport Properties of the One-Dimensional Heisenberg Chain： Bethe-Ansatz Solution with Twist Boundary Conditions

Based on the Bethe-Ansatz solution of the one-dimensional Heisenberg model under twist boundary conditions, we study the spectra of the persistent current carried by the low-lying excited states. It is shown that though the energy spectra of spin-singlet and spin-triplet excitations are degenerate, their persistent current spectra are quite different.     

Quantum statistics of the degenerate four-wave mixing process

A quantum mechanical treatment is given of the degenerate four-wave mixing process. The approximate Heisenberg equation of motion for creation and annihilation operators is solved taking into account attenuation and noise. The solution and the statistics of the modes are discussed both below and above the threshold condition.     

Convergence to equilibrium of the stochastic Heisenberg model

The stochastic Heisenberg model is a probabilistic model of the time evolution of a classical Heisenberg ferromagnet. It is proved that the stochastic process converges to equilibrium at sufficiently high temperatures, and that the equilibrium state is a Gibbs state of the Hamiltonian possessing the global Markov property. The principal technique employed is the expansion of observables on the state space into Laplace series.