An Operator Method of the Theory of Optical Coherence and Radiometry

This paper presents a quantum mechanical formalism of the classical coherence theory, within which the generalized radiance function defined in the time domain is regarded as a phase space representative of a time-dependent correlation operator of a polychromatic field. The theory deals with both stationary and nonstationary fields and, for a stationary field, provides a new operator formalism of the usual theory of optical coherence developed in the space-frequency domain. New results include an operator representation of the mutual coherence function, an operator version of the Wiener-Khintchine theorem, and an operator theorem that projects the correlation operator of a polychromatic field onto a particular spectral component. As illustrative examples, the previous formulas regarding the relationship between temporal coherence and spatial coherence, and the relationship between spectral properties and coherence properties are derived from the new operator formulas. The correspondence of the present formalism to the usual formalism using Dirac notation to describe the propagation of a stationary, partially coherent, quasi-monochromatic field is also considered.     

A System of Partial Differential Equations with High-Frequency Coefficients and Stokes Operator in the Main Part. Asymptotic Integration in the Case of Multiple Degeneration

The problem of time-periodic solutions is considered for a system of partial differential equations with the Stokes operator in the main part with coefficients rapidly oscillating by time in the case of multiple degeneration of the averaged stationary operator. The results concerning the existence and uniqueness of these solutions are proved and their asymptotic expansions are constructed and justified.     

Is gravitational mass of a composite quantum body equivalent to its energy?

We define passive gravitational mass operator of a hydrogen atom in the post-Newtonian approximation of general relativity and show that it does not commute with energy operator, taken in the absence of gravitational field. Nevertheless, the equivalence between the expectation values of passive gravitational mass and energy is shown to survive for stationary quantum states. Inequivalence between passive gravitational mass and energy at a macroscopic level results in time dependent oscillations of the expectation values of passive gravitational mass for superpositions of stationary quantum states, where the equivalence restores after averaging over time. Inequivalence between gravitational mass and energy at a microscopic level reveals itself as unusual electromagnetic radiation, emitted by the atoms, supported and moved in the Earth gravitational field with constant velocity using spacecraft or satellite, which can be experimentally measured.     

Some general problems of fast electrons transport

A generalization is given of the segments method in the form of a multistep method with generalized time for computing the transport of fast particles. The integral equation for a flow with generalized time in the phase space of variables is written under the assumption that the flow cuts the generalized time surface at right angles. The Green's function for the differential flow operator is the kernel of the integral equation. It is also shown that such an integral equation which can be obtained from a nonstationary kinetic equation provides a uniform consistent algorithm for solving either nonstationary or stationary problems. Examples of Green's functions are given for an operator of differential flow of fast electrons.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 110–114, August, 1974.The author would like to express his thanks to A. A. Vorob'ev and B. A. Kononov for their encouragement, to A. P. Yalovets for discussing the work with him, and to A. M. Kol'chuzhkin for going through the text.     

On an interpolation model for the transition operator for Markov and non-Markov processes

A phenomenological interpolation model for the transition operator of a stationary Markov process is shown to be equivalent to the simplest difference approximation in the master equation for the conditional density. Comparison with the formal solution of the Fokker-Planck equation yields a criterion for the choice of the correlation time in the approximate solution. The interpolation model is shown to be form-invariant under an iteration-cum-rescaling scheme. Next, we go beyond Markov processes to find the effective time-development operator (the counterpart of the conditional density) in the following very general situation: the stochastic interruption of the systematic evolution of a variable by an arbitrary stationary sequence of randomizing pulses. Continuous-time random walk theory with a distinct first-waiting-time distribution is used, along with the interpolation model for the transition operator, to obtain the solution. Convenient closed-form expressions for the ‘averaged’ time-development operator and the autocorrelation function are presented in various special cases. These include (i) no systematic evolution, but correlated pulses; (ii) systematic evolution interrupted by an uncorrelated (Poisson) sequence of pulses.     

延迟基因开关体系的稳定性及转变动力学

对延迟基因开关体系的稳定性及转变动力学进行了系统的研究．通过前向流采样及延迟随机模拟方法,获得了相空间稳态分布、转变路径和速率以及转变态系综的分布等信息．数值结果表明,基因开关体系中的延迟会减小稳态之间的间距,增大体系在转变态的布居．此外,转变速率也会随着延迟时间的增大单调增加,这些现象都说明延迟会削弱双稳基因开关体系的稳定性．转变路径由两种转录因子蛋白质的总的粒子数的差和操纵子位点的状态来描述,对转变路径的数值分析表明转录和翻译过程导致的延迟会显著影响转变动力学．特别是对于转变态系综而言,操纵子结合两种不同二聚体概率的差异随着延迟时间的增加逐渐加大.     

Pseudo-Hermiticity and electromagnetic wave propagation: The case of anisotropic and lossy media

Pseudo-Hermitian operators can be used in modeling electromagnetic wave propagation in stationary lossless media. We extend this method to a class of non-dispersive anisotropic media that may display loss or gain. We explore three concrete models to demonstrate the utility of our general results and reveal the physical meaning of pseudo-Hermiticity and quasi-Hermiticity of the relevant wave operator. In particular, we consider a uniaxial model where this operator is not diagonalizable. This implies left-handedness of the medium in the sense that only clockwise circularly polarized plane-wave solutions are bounded functions of time.     

On the stationary properties of an open system with internal coherent interactions

The asymptotic behaviour of an open system in which coherent and incoherent interactions interfere is discussed. It is shown that a stationary density operator exists which is diagonal with the coherent interaction, and that detailed balance is satisfied. A model-independent form for this density operator is given. A class of systems is identified for which the stationary solution can be constructed explicitly and in which some known models are contained.     

Relaxation of physical systems in relative-number state representation

The relaxation properties of physical systems in the Liouville space are investigated in terms of the relative-number state representation. An arbitrary state can be expressed by superposition of relative-number states. In the absence of an time-dependent external field, all components with non-zero relative-numbers decay to zero with time, and any stationary state can be expressed only in terms of zero relative-number states. The phase canonically conjugate to the relative-number is completely uncertain in a stationary state. It is thought that relaxation from an arbitrary initial state to a stationary state is described as some kind of phase relaxation process. Such a phase relaxation process is explicitly described by the phase operator formalism within the framework of the relative-number state representation.     

Exact Green's function of the equations of quantum mechanics in an electromagnetic field. I

An exact Green's function of the Cauchy problem in arbitrary (nonparallel) stationary homogeneous electrical and magnetic fields is constructed for the Klein-Gordon-Fock and Dirac equations and for the Pauli equation in arbitrary nonstationary fields, on the basis of a single approach using the method of the canonical Maslov operator (extension of the WKB method to the multidimensional case) and a Fock idea about the proper time in relativitic mechanics.     

Quantum effects of periodic orbits for the kicked top

We analyze traces of powers of the time evolution operator of a periodically kicked top. Semiclassically, such traces are related to periodic orbits of the classical map. We derive the semiclassical traces in a coherent state basis and show how the periodic orbits can be recovered via a Fourier transform. A breakdown of the stationary phase approximation is detected. The quasi energy spectrum remains elusive due to lack of knowledge of sufficiently many periodic orbits. Divergencies of periodic orbit formulas are avoided by appealing to the finiteness of the quantum mechanical Hilbert space. The traces also enter the coefficients of the characteristic polynominal of the Floquet operator. Statistical properties of these coefficients give rise to a new criterion for the distinction of chaos and regular motion.     

The Darboux Transform for the One-Dimensional Stationary Dirac Equation

Based on the method oftransformation operators, the Darboux transformation operator has been constructed for the one-dimensional stationary Dirac equation. The properties of this operator have been studied. As an application, exactly solvable transparent potentials and potentials with the spectrum of a relativistic harmonic oscillator have been obtained.     

Scale-covariant quantization

It is argued that conventional operator field equations derived from a stationary action principle are improper when local operator products involve infinite multiplicative renormalizations. An alternative, scale-covariant operator field equation is derived by making scale variations of the field, variations which are everywhere proportional to the field itself. The resultant equation of motion is the same as that previously found on the basis of augmented field theory.     

A Thermodynamic Formalism for Continuous Time Markov Chains with Values on the Bernoulli Space: Entropy, Pressure and Large Deviations

Through this paper we analyze the ergodic properties of continuous time Markov chains with values on the one-dimensional spin lattice $\{1,\dots,d\}^{{\mathbb{N}}}$ (also known as the Bernoulli space). Initially, we consider as the infinitesimal generator the operator , where is a discrete time Ruelle operator (transfer operator), and $A:\{1,\dots,d\}^{{\mathbb{N}}}\to\mathbb{R}$ is a given fixed Lipschitz function. The associated continuous time stationary Markov chain will define the a priori probability. Given a Lipschitz interaction $V:\{1,\dots,d\}^{{\mathbb{N}}}\to\mathbb{R}$ , we are interested in Gibbs (equilibrium) state for such V. This will be another continuous time stationary Markov chain. In order to analyze this problem we will use a continuous time Ruelle operator (transfer operator) naturally associated to V. Among other things we will show that a continuous time Perron-Frobenius Theorem is true in the case V is a Lipschitz function. We also introduce an entropy, which is negative (see also Lopes et al. in Entropy and Variational Principle for one-dimensional Lattice Systems with a general a-priori probability: positive and zero temperature. Arxiv, 2012), and we consider a variational principle of pressure. Finally, we analyze large deviations properties for the empirical measure in the continuous time setting using results by Y. Kifer (Tamsui Oxf. J. Manag. Sci. 321(2):505–524, 1990). In the last appendix of the paper we explain why the techniques we develop here have the capability to be applied to the analysis of convergence of a certain version of the Metropolis algorithm.     

Calculation of MAS spectra influenced by slow molecular tumbling.

A numeric algorithm is proposed that is suitable to calculate spectral lineshapes influenced by isotropic and anisotropic tumbling under sample spinning conditions. It is based on the stochastic Liouville equation and a rotational diffusion process described by a stationary Markov operator. A corresponding FORTRAN program can be implemented on a regular personal computer. The calculations result in spectral lineshapes including a complete set of spinning sidebands. The sensitive time scale of the resulting lineshapes depends on the deviation of the sample spinning axis from the magic angle. An example is presented demonstrating the potential of off-magic-angle spinning as a tool to analyze slow tumbling motions.     

Stationary momentum space solution of the Fokker-Planck equation for a simple model of a laser oscillator exhibiting spatial dispersion

Starting from a multimode hamiltonian for a system of radiating oscillators coupled with atomic reservoirs, the secular master equation for the radiation-density operator is calculated in the interaction picture after elimination of the atomic variables. Using the differential operator representation for coherent states this equation is transcribed into a multimode Fokker-Planck equation. The stationary solution in momentum space is given for the threshold region. Fourier transformation to configuration space results in a quasi-free energy formula for a laser oscillator exhibiting spatial dispersion.     

Disordered quantum walks in two-dimensional lattices

The properties of the two-dimensional quantum walk with point,line,and circle disorders in phase are reported.Localization is observed in the two-dimensional quantum walk with certain phase disorder and specific initial coin states.We give an explanation of the localization behavior via the localized stationary states of the unitary operator of the walker+coin system and the overlap between the initial state of the whole system and the localized stationary states.     

The numerical operator method to the real time dynamics of currents through the nanostructures with different topologies

We present the numerical operator method for studying the transient and stationary currents through nanostructures with different topologies, e.g., the flakes of square and honeycomb lattices. We find a quasi-stationary stage in the transient currents through the square flakes, but none in the honeycomb flakes. The stationary current through the square flakes increases smoothly with the voltage bias. In contrast, there is a threshold voltage in the current-voltage curve through a honeycomb flake, indicating a gap at the Fermi energy.     

On the quantum mechanical description of scattering by a dissipative and diffusive scattering centre

A partial integro-differential equation is formulated for the Wigner transform of the quantum mechanical reduced density operator describing the time evolution of a “macroscopic” coordinate under the influence of coupling to a large number of “intrinsic” degrees of freedom. The equation contains integral operators which lead to energy dissipation and diffusion and reduces to a transport equation of the Fokker-Planck type if the form factors in the integrands are treated in appropriate (harmonic) approximations. The stationary solution of the partial integro-differential equation is obtained numerically for scattering by a conservative potential and by a dissipative and diffusive scattering centre in one spatial dimension.     

Note on the symmetrized density operator treatment of quantized dissipative systems

An anti-symmetrization procedure will be introduced for the zero-point term in the symmetrized density operator equation for the quantized analogue of a simple classical order parameter equation. An explicit solution is given for the Wigner-distribution in the stationary state. The present approach extends and corrects a previous treatment.