Stochastic Processes and Mean Field Systems Defined by Nonlinear Markov Chains: An Illustration for a Model of Evolutionary Population Dynamics

In physics, there is a growing interest in studying stochastic processes described by evolution equations such as nonlinear master equations and nonlinear Fokker–Planck equations that define the so-called nonlinear Markov processes and are nonlinear with respect to probability densities. In this context, however, relatively little is known about nonlinear Markov processes defined by nonlinear Markov chains. In the present work, we demonstrate explicitly how the nonlinear Markov chain approach can be carried out by addressing a model for evolutionary population dynamics. In line with the nonlinear Markov chain approach, we derive a measure that tells us how attractive it is for a biological entity to evolve towards a particular biological type. Likewise, a measure for the noise level of the evolutionary process is obtained. Both measures are found to be implicitly time dependent. Finally, a simulation scheme for the many-body system corresponding to the Markov chain model is discussed.     

Wave processes in media with hysteretic nonlinearity: Part 2

Nonlinear processes caused by the propagation of low-frequency and high-frequency acoustic pulses in an unbounded medium and the propagation of continuous waves in a ring resonator are theoretically studied on the basis of two hysteretic equations of state for media with imperfect elasticity. The profiles and parameters of pulses, the resonance curve and the Q factor of the resonator, and the ratio of the nonlinear resonance frequency shift to the nonlinear damping decrement are determined. For nonlinear wave processes in such media, the distinctive features that allow one to choose an appropriate hysteretic equation of state for analytically describing the experimental data are revealed.     

On the generation of picosecond light continua depending on various nonlinear processes

The generation of picosecond light continua depends on various nonlinear processes such as parametric four-photon interaction, self-phase modulation, self-focusing, and avalanche ionization. In order to systematize the influence of these processes we have indicated both the threshold for continuum generation measured by various authors and the threshold for the nonlinear processes in a power diameter diagram characterizing the laser beam. The result is that all the thresholds for continuum measured under very different experimental conditions are situated nearly on one curve given by the nonlinear process with the lowest threshold. We have carried out measurements on the spectral energy distribution and on the angular distribution of the continuum generated in water and alkali halides by a mode-locked single-pulse Nd-glass laser. Under our experimental conditions both self-phase modulation and parametric four-photon interaction contribute to continuum generation.     

Nonlinear photonic crystals: II. Interaction classification for quadratic nonlinearities

Weakly nonlinear interactions between wavepackets in lossless periodic dielectric media are studied based on the classical nonlinear Maxwell equations. We consider nonlinear processes such that: (i) the amplitude of the wave component due to the nonlinearity does not exceed the amplitude of its linear component; (ii) the spatial range of a probing wavepacket is much smaller than the dimension of the medium sample, and it is not too small compared with the dimension of the primitive cell. These nonlinear processes are naturally described in terms of the Bloch modes and the dispersion relations of the underlying linear periodic medium. It turns out that only a few triads of modes have significant nonlinear interactions. They are singled out by the frequency and phase matching conditions and, as we show, by an additional selection rule: the group velocity matching condition. The latter condition is the most important selection rule for the nonlinear regimes. We give a complete quantitative classification of all possible significant interactions for quadratic nonlinearities. The classification is based on a universal system of indices reflecting the intensity of nonlinear interactions. The obtained classification points to the second harmonic generation as being one of the stronger nonlinear interactions, and often the strongest one.     

Nonlinear wave interaction in photonic band gap materials

We present detailed analytical and numerical studies of nonlinear wave interaction processes in one-dimensional (1D) photonic band gap (PBG) materials with a Kerr nonlinearity. We demonstrate that some of these processes provide efficient mechanisms for dynamically controlling so-called gap-solitons. We derive analytical expressions that accurately determine the phase shifts experienced by nonlinear waves for a large class of non-resonant interaction processes. We also present comprehensive numerical studies of inelastic interactions, and show that rather distinct regimes of interaction exist. The predicted effects should be experimentally observable, and can be utilized for probing the existence and parameters of gap solitons. Our results are directly applicable to other nonlinear periodic structures such as Bose–Einstein condensates in optical lattices.     

Nonlinear photonic crystals: II. Interaction classification for quadratic nonlinearities

Abstract

Weakly nonlinear interactions between wavepackets in lossless periodic dielectric media are studied based on the classical nonlinear Maxwell equations. We consider nonlinear processes such that: (i) the amplitude of the wave component due to the nonlinearity does not exceed the amplitude of its linear component; (ii) the spatial range of a probing wavepacket is much smaller than the dimension of the medium sample, and it is not too small compared with the dimension of the primitive cell. These nonlinear processes are naturally described in terms of the Bloch modes and the dispersion relations of the underlying linear periodic medium. It turns out that only a few triads of modes have significant nonlinear interactions. They are singled out by the frequency and phase matching conditions and, as we show, by an additional selection rule: the group velocity matching condition. The latter condition is the most important selection rule for the nonlinear regimes. We give a complete quantitative classification of all possible significant interactions for quadratic nonlinearities. The classification is based on a universal system of indices reflecting the intensity of nonlinear interactions. The obtained classification points to the second harmonic generation as being one of the stronger nonlinear interactions, and often the strongest one.     

Nonlinear control of the propagation of optical pulses in doped optical fibers

Results of study of nonlinear processes occurring during the propagation of light pulses in optical fibers doped with atoms of rare-earth elements under conditions of atomic coherence and interference are presented. For a three-level Λ scheme of interaction, the linear (χ⁽¹⁾) and nonlinear (χ⁽³⁾) susceptibilities of such a medium are calculated. It is shown that the coefficients of Kerr nonlinearity and nonlinear absorption can reach extremely large values and can be negative. The competition between linear and nonlinear processes in the Λ scheme allows one to obtain compensation regimes when the coefficients of dispersion or absorption of the optical fiber material vanish. The efficient control of the optical properties of such a system over wide limits proves to be possible owing to variations in the parameters of light pulses at the input of the medium. The necessary conditions for realizing regimes with “slow” light, as well as for self-compression of a probing pulse on ultimately small spatial scales in a doped optical fiber, were obtained.     

Distinction of the competing processes in nonlinear resonant frequency mixing in atoms

A coherent transient response technique is applied, for the first time, to distinguish between the influence of competing processes such as saturation of atomic transition and effects due to laser-induced nonlinear polarization on nonlinear resonant mixing. Experimental research is performed for the 3s–5s two-photon transition in sodium vapour. Delaying the probe picosecond pulse after the exciting one, variations of efficiency and phase-matched ratio upon incident intensity and exciting wavelength are investigated.     

Wave processes in media with hysteretic nonlinearity. Part I

Two phenomenological models of hysteretic equations of state for media with imperfect elasticity are described and compared. On the basis of these equations, a theoretical study of nonlinear effects caused by the acoustic wave propagation in an unbounded medium is performed. The profiles, parameters, and spectra of waves are determined. The distinctive features of nonlinear wave processes in such media are revealed, so that these features can be used to choose the appropriate hysteretic equation of state for analytically describing the experimental data.     

Role of the longitudinal piston error in a tiled-grating compressor in second and high-order harmonic generation

In this work, we study the limitations for overcoming the longitudinal piston error in a femtosecond tiled-grating compressor using nonlinear measurements like second harmonic generation. In particular, we observe the influence of this error when developing high-power laser experiments such as high-order harmonic generation. The generation of nonlinear processes with femtosecond pulses compressed in tiled-grating systems is studied. Special attention is paid to the role of the longitudinal piston error which is the most difficult to overcome in the compressor alignment. A complex spatio-temporal structure is expected to appear due to that misalignment. Both second harmonic generation in nonlinear crystals and high-order harmonic generation in gases are studied and a strong dependence with piston error is found, thus leading to a sub-micron modulation in the generated signal. In particular, the high sensitivity of the high-order harmonics to the longitudinal position allows one to use this processes for the accurate alignment of the compressor to few tens of nanometers.     

Self-induced transparency and self-induced darkening with a nonlinear frequency-doubling polarization interferometer

It is shown that the system polarizer-frequency-doubling crystal(s)-analyzer has a nonlinear transmission for the fundamental wave. The intensity-dependent transmission of this device is due to the nonlinear phase shift that the fundamental beam obtains in the nonlinear crystal as a result of cascaded second-order processes. Depending on the mutual orientation of the polarizer and the analyzer such effects as self-induced transparency and self-induced darkening can be realized.     

Extreme quantum field theory and particle physics with IZEST

The prospect of next-generation ultra-high-intensity laser sources has prompted recent renewed study of nonlinear QED processes, such as the Schwinger effect, in which the instability of the QED vacuum is probed by external fields. Experimental observation of these nonlinear QED effects would provide unprecedented controlled access to non-perturbative processes in quantum field theory under extreme conditions, which is of direct interest in particle physics and astrophysical applications. I summarize important theoretical issues, both conceptual and computational, related to these nonlinear QED effects.     

求解非线性Poisson方程的格子演化算法

非线性Poisson方程在化学、化工及生物等领域有着广泛的应用。本文发展了一种基于格子演化的新算法-格子Poisson方法(LPM),并且给出了Dirichlet边界条件和Neumann边界条件的实现方法。本方法不需要对方程进行线化处理,直接求解非线性方程,适用范围广泛。Dirichlet边界与Neumann边界的数值模拟结果与多重网格法等结果符合很好,验证了该方法在求解非线性Poisson方程的正确性与有效性。本方法非常适合并行计算,并方便扩展到三维情况。     

Wavelength tuning of femtosecond pulses generated in nonlinear crystals by using diffractive lenses

We demonstrate that diffractive lenses (DLs) can be used as a simple method to tune the central wavelength of femtosecond pulses generated from second-order nonlinear optical processes in birefringent crystals. The wavelength tunability is achieved by changing the relative distance between the nonlinear crystal and the DL, which acts in a focusing configuration. Besides the many practical applications of the so-generated pulses, the proposed method might be extended to other wavelength ranges by demonstrated similar effects on other nonlinear processes, such as high-order harmonic generation.     

Variation of Parameters Method for Solving System of Nonlinear Volterra Integro-Differential Equation

It is well known that nonlinear integro-differential equations play vital role in modeling of many physical processes, such as nano-hydrodynamics, drop wise condensation, oceanography, earthquake and wind ripple in desert. Inspired and motivated by these facts, we use the variation of parameters method for solving system of nonlinear Volterra integro-differential equations. The proposed technique is applied without any discretization, perturbation, transformation, restrictive assumptions and is free from Adomian's polynomials. Several examples are given to verify the reliability and efficiency of the proposed technique.     

Generation of compound non-gaussian random processes with a given correlation function

A compound representation of random processes is considered. Each independent component of such a process is considered as the solution of the proper stochastic differential equation (SDE). This guarantees that the process obtained is stationary and ergodic. The analytical expressions are developed for nonlinear coefficients of the generating SDE. Theoretical results are compared with numerical simulation.     

Generation of Competitive and Coexisting Two Pairs of Biphotons in Single Hot Atomic Vapor Cell

The non-classical states of light serve as a potential candidate for emerging quantum information process. A processing trend to enhance its scalability is to integrate multiple nonlinear processes with the dressed-state picture into a single device, and therefore, are useful for quantum computations. Here, a novel method is proposed to experimentally achieve the generation of coexisting two pairs of narrow-band biphotons by two four-wave mixing processes in a single hot rubidium vapor cell. Based on the photon-atom nonlinear interfaces, the generated biphotons exhibits genuine entanglement in time–energy. Meanwhile, the nonlinear susceptibility with the dressed-state picture decides the temporal correlation of the biphotons wave packet as a damped periodic Rabi oscillation, suggesting the property of the high-dimensional time–energy entangled state. Such a high-dimensional entangled state also is an efficient way to enhance information carrying capacity. By alternating two nonlinear susceptibilities in a single device, respectively, there exists a competition of the generation rate between such two pairs of biphotons. Moreover, both generated two pairs of biphotons that violate the Cauchy–Schwarz inequality and show their non-classical behavior.     

Asymptotic solutions of Fokker-Planck equations for nonlinear irreversible processes and generalized Onsager-Machlup theory

Fokker-Planck equations for nonlinear processes are solved asymptotically in the limit k→0 where k is the Boltzmann constant. It is shown that the leading asymptotic solutions for conditional (two-gate) distribution functions simply correspond to generalizations of the Onsager-Machlup theory to nonlinear processes. The asumptotic solution method used in the paper is similar to the well-known W.K.B. method in quantum mechanics. A stability criterion of nonlinear irreversible processes is also considered and compared with the Glansdorff-Prigogine stability criterion.     

Modulation instability: The beginning

We discuss the early history of an important field of “sturm and drang” in modern theory of nonlinear waves. It is demonstrated how scientific demand resulted in independent and almost simultaneous publications by many different authors on modulation instability, a phenomenon resulting in a variety of nonlinear processes such as envelope solitons, envelope shocks, freak waves, etc. Examples from water wave hydrodynamics, electrodynamics, nonlinear optics, and convection theory are given.     

Adiabatic processes in frequency conversion

Adiabatic evolution, an important dynamical process in a variety of classical and quantum systems providing a robust way of steering a system into a desired state 1 - 5 , was introduced only recently to frequency conversion 6 - 9 . Adiabatic frequency conversion allowed the achievement of efficient scalable broadband frequency conversion 8 , 9 and was applied successfully to the conversion of ultrashort pulses, demonstrating near‐100% efficiency for ultrabroadband spectrum 10 - 16 . The underlying analogy between undepleted pump nonlinear processes and coherently excited quantum systems was extended in the past few years to multi‐level quantum systems, demonstrating new concepts in frequency conversion, such as complete frequency conversion through an absorption band 17 - 20 . Additionally, the undepleted pump restriction was removed, enabling the exploration of adiabatic processes in the fully nonlinear dynamics regime of nonlinear optics 21 - 26 . In this article, the basic concept of adiabatic frequency conversion is introduced, and recent advances in ultrashort physics, multi‐process systems, and the fully nonlinear dynamics regime are reviewed.