Galilean-Covariant Clifford Algebras in the Phase-Space Representation

We apply the Galilean covariant formulation of quantum dynamics to derive the phase-space representation of the Pauli–Schrödinger equation for the density matrix of spin-1/2 particles in the presence of an electromagnetic field. The Liouville operator for the particle with spin follows from using the Wigner–Moyal transformation and a suitable Clifford algebra constructed on the phase space of a (4 + 1)-dimensional space–time with Galilean geometry. Connections with the algebraic formalism of thermofield dynamics are also investigated.     

Anisotropic relaxation and motion of half-integer quadrupole nuclei studied by central transition nuclear magnetic resonance spectroscopy

An efficient theoretical formalism and advanced experimental methods are presented for studying the effects of anisotropic molecular motion and relaxation on solid-state central transition NMR spectra of half-integer quadrupole nuclei. The theoretical formalism is based on density operator algebra and involves the stochastic Liouville–von Neumann equation. In this approach the nuclear spin interactions are represented by the Hamiltonian while the motion is described by a discrete stochastic operator. The nuclear spin interactions fluctuate randomly in the presence of molecular motion. These fluctuations may stimulate the relaxation of the system and are represented by a discrete relaxation operator. This is derived from second-order perturbation theory and involves the spectral densities of the system. Although the relaxation operator is valid only for small time intervals it may be used recursively to obtain the density operator at any time. The spectral densities are allowed to be explicitly time dependent making the approach valid for all motional regimes. The formalism has been applied to simulate partially relaxed central transition ¹⁷O NMR spectra of representative model systems. The results have revealed that partially relaxed central transition lineshapes are defined not only by the nuclear spin interactions but also by anisotropic motion and relaxation. This has formed the basis for the development of central transition spin-echo and inversion-recovery NMR experiments for investigating molecular motion in solids. As an example we have acquired central transition spin-echo and inversion-recovery ¹⁷O NMR spectra of polycrystalline cristobalite (SiO₂) at temperatures both below and above the α–β phase transition. It is found that the oxygen atoms exhibit slow motion in α-cristobalite. This motion has no significant effects on the fully relaxed lineshapes but may be monitored by studying the partially relaxed spectra. The α–β phase transition is characterized by structural and motional changes involving a slight increase in the Si–O–Si bond angle and a substantial increase in the mobility of the oxygen atoms. The increase in the Si–O–Si angle is supported by the results of ¹⁷O and ²⁹Si NMR spectroscopy. The oxygen motion is shown to be orders of magnitude faster in β-cristobalite resulting in much faster relaxation and characteristic lineshapes. The measured oscillation frequencies are consistent with the rigid unit mode model. This shows that solid-state NMR and lattice dynamics simulations agree and may be used in combination to provide more detailed models of solid materials.     

Phase Properties of New Even and Odd Nonlinear Coherent States

Using the Pegg–Barnett formalism of phase operator, we obtain phase probability distributions of new even and odd nonlinear coherent states. It is shown that the distributions for the states are rather different, and unlike the case of ordinary even and odd coherent states the Pegg–Barnett distribution clearly reflects the different character of quantum interference in the case of the new even and odd coherent states.     

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set t₀=t is naturally understood where t₀ is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator A in the evolution equation is semi-simple, i.e., diagonalizable; when A is not semi-simple and has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be Polchinski's theorem for renormalizable field theories. We apply the method to interface dynamics such as kink–anti-kink and soliton–soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.     

Time-domain master equation for pulse evolution and laser mode-locking

Using the well-known analogy between the space and time domains we derive a temporal master equation (ME) operator which can be applied to any cavity containing dispersive and filtering elements, phase or amplitude modulators, and one nonlinear element. The cavity properties are described in terms of 2 × 2 `KIJL' matrices. We show that this ME correctly reproduces the cavity mode structure in the linear limit. Numerical simulation of an actively mode-locked Fabry–Perot laser with the nonlinear medium at an end mirror gives results in excellent agreement with those found using the more conventional Huygens' integral method. Using a simple perturbation approach based on the nonlinear Schrödinger equation (NLS) we also show that the field in this laser is soliton-like, and give analytic expressions for the soliton parameters.     

Effective Hamiltonians and Phase Diagrams for Tight-Binding Models

We present rigorous results for several variants of the Hubbard model in the strong-coupling regime. We establish a mathematically controlled perturbation expansion which shows how previously proposed effective interactions are, in fact, leading-order terms of well-defined (volume-independent) unitarily equivalent interactions. In addition, in the very asymmetric (Falicov–Kimball) regime, we are able to apply recently developed phase-diagram technology (quantum Pirogov–Sinai theory) to conclude that the zero-temperature phase diagrams obtained for the leading classical part remain valid, except for thin excluded regions and small deformations, for the full-fledged quantum interaction at zero or low temperature. Moreover, the phase diagram is stable against addition of arbitrary, but sufficiently small further quantum terms that do not break the ground-state symmetries. This generalizes and unifies a number of previous results on the subject; in particular, published results on the zero-temperature phase diagram of the Falikov–Kimball model (with and without magnetic flux) are extended to small temperatures and/or small ionic hopping. We give explicit expressions for the first few orders, in the hopping amplitude, of equivalent interactions, and we describe the resulting phase diagram. Our approach yields algorithms to compute equivalent interactions to arbitrarily high order in the hopping amplitude.     

Perturbation calculation for quasi-equilibrium statistical operator

A perturbation calculation is applied to the equation of motion for the quasi-equilibrium statistical operator. A special case of the time evolution of the quasi-equilibrium statistical operator inclusive of the second perturbation order is considered.     

SENSITIVITY TO PERTURBATION IN QUANTUM CHAOTIC SYSTEM

Numerical results show that, for quantum autonomous chaotic system, the evolution of initially coherent states are sensitive to perturbation. The overlap of a perturbed state with the unperturbed one decays exponentially, which is followed by fluctuation around N^-1, N being the dimension of the Hilbert space. The matrix elements of the evolution operator in interaction picture tend to be a random distribution after sufficiently long time, where the interaction is the perturbation, even when the perturbation is very weak. The difference between a regular system and the chaotic one is shown clearly. In a regular system, the overlap shows strong revival. The distribution of the evolution matrix has only a few dominant terms.     

A lattice test of strong coupling behaviour in QCD at finite temperature

We propose a set of lattice measurements which could test whether the deconfined, quark–gluon plasma, phase of QCD shows strong coupling aspects at temperatures a few times the critical temperature for deconfinement. The measurements refer to twist-two operators which are not protected by symmetries and which in a strong-coupling scenario would develop large, negative, anomalous dimensions, resulting in a strong suppression of the respective lattice expectation values in the continuum limit. Special emphasis is put on the twist-two operator with lowest spin (the spin-2 operator orthogonal to the energy–momentum tensor within the renormalization flow) and on the case of quenched QCD, where this operator is known for arbitrary values of the coupling: this is the quark energy–momentum tensor. The proposed lattice measurements could also test whether the plasma constituents are pointlike (as expected at weak coupling), or not.     

Characterization of Stochastic Bifurcations in a Simple Biological Oscillator

This study of the effect of noise on bifurcations in a simple biological oscillator with a periodically modulated threshold uses the first-passage-time problem of the Ornstein–Uhlenbeck process with a periodic boundary to define the operator governing the transition of a threshold phase density. Stochastic phase-locking is analyzed numerically by evaluating the evolution of the probability density function of the threshold phase. A firing phase map in a noisy environment is extended to a stochastic kernel so that stochastic bifurcations can be investigated by spectral analysis of the kernel.     

Numerical simulation of soliton and kink density waves in traffic flow with periodic boundaries

The soliton and kink–antikink density waves are simulated with periodic boundaries, by adding perturbation in the initial condition on single-lane road based on a car-following model. They are reproduced in the form of the space–time evolution of headway, both of which propagate backwards. It is found that the solitons appear only near the neutral stability line regardless of the boundary conditions, and they exhibit upward form when the initial headway is smaller than the safety distance, otherwise they exhibit downward form. Comparison is made between the numerical and analytical results about the amplitude of kink–antikink wave, and the underlying mechanism is analyzed. Besides, it is indicated that the maximal current of traffic flow increases with decreasing safety distance. The numerical simulation shows a good agreement with the analytical results.     

Global Phase Time and Wave Function for the Kantowski-Sachs Anisotropic Universe

A consistent quantization with a clear notion of time and evolution is given for the anisotropic Kantowski–Sachs cosmological model. It is shown that a suitable coordinate choice allows to obtain a solution of the Wheeler–DeWitt equation in the form of definite energy states, and that the results can be associated to two disjoint equivalent theories, one for each sheet of the constraint surface.     

Stochastic phase lockings in a relaxation oscillator forced by a periodic input with additive noise: A first-passage-time approach

Noise effects on phase lockings in a system consisting of a piecewise-linear van der Pol relaxation oscillator driven by a periodic input are studied. The problem of finding the period of the oscillator is reduced to the first-passage-time problem of the Ornstein-Uhlenbeck process with time-varying boundary. The probability density functions of the first-passage time are used to define the operator which governs a transition of an input phase density after one cycle of the oscillator. Phase lockings in a stochastic sense are investigated on the basis of the density evolution by the operator.     

The Phase Transition in Statistical Models Defined on Farey Fractions

We consider several statistical models defined on the Farey fractions. Two of these models may be regarded as spin chains, with long-range interactions, while another arises in the study of multifractals associated with chaotic maps exhibiting intermittency. We prove that these models all have the same free energy. Their thermodynamic behavior is determined by the spectrum of the transfer operator (Ruelle–Perron–Frobenius operator), which is defined using the maps (presentation functions) generating the Farey tree. The spectrum of this operator was completely determined by Prellberg. It follows that these models have a second-order phase transition with a specific heat divergence of the form C [ ln² ]^–1. The spin chain models are also rigorously known to have a discontinuity in the magnetization at the phase transition.     

Squeezed States and Nondiagonal P-Representation

The R-representation and the nondiagonalP-representation for density operator withsqueezed-state basis are defined. The special cases forchaotic and laser fields are calculated. TheFokker–Planck equation for the damped harmonic oscillator withsqueezed bath is considered and the steady-statesolution is given. A special case of the steady-statesolution for only a thermal bath is shown. ThePegg–Barnett phase distribution is compared with the radialintegration on the generalized P-function.     

Spectral Analysis of Stochastic Phase Lockings and Stochastic Bifurcations in the Sinusoidally Forced van der Pol Oscillator with Additive Noise

Noise effects on the phase lockings and bifurcations in the sinusoidally forced van der Pol relaxation oscillator are investigated. Deterministic (noise-free) one-dimensional Poincaré mapping is extended to the iteration of the operator defined by a stochastic kernel function. Stochastic phase lockings and bifurcations are analyzed in terms of the density evolution by the operator. In particular, a new method which uses spectra (eigenvalues and eigenfunctions) of the operator to analyze stochastic bifurcations intensively is proposed.     

Large Time Dynamics of a Classical System Subject to a Fast Varying Force

We investigate the asymptotic behavior of solutions to a kinetic equation describing the evolution of particles subject to the sum of a fixed, confining, Hamiltonian, and a small, time-oscillating, perturbation. The equation also involves an interaction operator which acts as a relaxation in the energy variable. This paper aims at providing a classical counterpart to the derivation of rate equations from the atomic Bloch equations. In the present classical setting, the homogenization procedure leads to a diffusion equation in the energy variable, rather than a rate equation, and the presence of the relaxation operator regularizes the limit process, leading to finite diffusion coefficients. The key assumption is that the time-oscillatory perturbation should have well-defined long time averages: our procedure includes general “ergodic” behaviors, amongst which periodic, or quasi-periodic potentials only are a particular case.     

光纤中扰动的小信号增益

从非线性薛定谔方程出发,在小信号近似下,推导并求解了光纤中扰动相位和幅度的演化方程,利用得到的扰动相位及功率增益的表达式,研究了初相位和频率对传输过程中扰动增益的影响。研究表明：扰动的初相位对扰动增益的初值和初始阶段的演化规律有重要影响;取决于扰动初相位,任何一个频率的扰动增益都有可能达到一个共同的最大值;在被认为无调制不稳定的正色散区和扰动频率大于截止频率的负色散区,扰动增益随距离是振荡的;在被认为有调制不稳定的扰动频率小于截止频率的负色散区,频率相同而初相位不同的扰动增益将经历不同形式的演化后趋于同一正值。     

Landau problem with a general time-dependent electric field

The time evolution is studied for the Landau level problem with a general time dependent electric field E(t) in a plane perpendicular to the magnetic field. A general and explicit factorization of the time evolution operator is obtained with each factor having a clear physical interpretation. The factorization consists of a geometric factor (path-ordered magnetic translation), a dynamical factor generated by the usual time-independent Landau Hamiltonian, and a nonadiabatic factor that determines the transition probabilities among the Landau levels. Since the path-ordered magnetic translation and the nonadiabatic factor are, up to completely determined numerical phase factors, just ordinary exponentials whose exponents are explicitly expressible in terms of the canonical variables, all of the factors in the factorization are explicitly constructed. New quantum interference effects are implied by this result. The factorization is unique from the point of view of the quantum adiabatic theorem and provides a seemingly first rigorous demonstration of how the quantum adiabatic theorem (incorporating the Berry phase phenomenon) is realized when infinitely degenerate energy levels are involved. Since the factorization separates the effect caused by the electric field into a geometric factor and a nonadiabatic factor, it makes possible to calculate the nonadiabatic transition probabilities near the adiabatic limit. A formula for matrix elements that determines the mixing of the Landau levels for a general, nonadiabatic evolution is also provided by the factorization.     

Second Order Perturbations of a Schwarzschild Black Hole: Inclusion of Odd Parity Perturbations

We consider perturbations of a Schwarzschild black hole that can be of both even and odd parity, keeping terms up to second order in perturbation theory, for the l = 2 axisymmetric case. We develop explicit formulae for the evolution equations and radiated energies and waveforms using the Regge–Wheeler–Zerilli approach. This formulation is useful, for instance, for the treatment in the "close limit approximation" of the collision of counterrotating black holes.