Level-crossing Problem Research Articles

The level-crossing problem of a weakly damped particle in quadratic potential well under harmonic excitation

Escape and level-crossing are fundamental and closely related problems in transient dynamics. Often, when a particle reaches a critical displacement, its escape becomes inevitable. Therefore, escape models based on truncated potentials are often used, resulting in similar problems to level-crossing formulations. Two different types of dynamics can be identified, leading to different kinds of level-crossing depending on the relationship between the damping and the excitation level. The first one (“fast escape”) is mainly governed by the initial energy of the system, which is determined through the initial conditions. The second one (“slow escape”) is governed by the beatings determined through the relationship between external excitation and damping. An analytic approach for estimating the size and location of the safe basins (SBs) in the plane of the initial conditions (ICs) of a 1-DOF externally excited oscillator is suggested. It enables the identification of the set of ICs where the particle never reaches a certain threshold under the given excitation. The SBs depend on the damping coefficient and the excitation’s amplitude, frequency, and phase. Nonetheless, one can describe the essential properties of an SBs in the case of the almost resonant excitation using only two parameters: the forced response amplitude and the damping coefficient ratio to the difference between the natural and the excitation frequencies. Although the analysis is performed for a linear oscillator, it provides insight into the rush erosion process of the SBs (“Dover cliff” phenomenon), described previously only for nonlinear systems. The analysis reveals that the “Dover cliff” phenomenon is related to the decay rate of the transient motion and that it can occur even in linear systems too. From the engineering point of view, the rush erosion of the SBs is critical in noisy environments where devices operating in regions close to the “Dover cliff” are unsafe. Due to its simplicity, the proposed mechanical model might be generic for further analysis of the escape and level-crossing problems considering various nonlinearities (e.g., Coulomb friction, small polynomial-type nonlinearities of the restoring force, or constant restoring force). Possible applications include but are not limited to avoiding collisions for systems with clearances and durability analysis of brittle materials subjected to noisy loads.

A Fourier approach to pulse pile-up in photon-counting x-ray detectors.

An analytic Fourier approach to predict the expected number of counts registered in a photon-counting detector subject to pulse pile-up for arbitrary photon flux, detector response function, and pulse-shape is presented. The analysis provides a complete forward model for energy-sensitive, photon-counting x-ray detectors for spectral computed tomography. The formalism of the stochastic theory of the expected frequency of level crossings of shot noise processes is applied to the pulse pile-up effect and build on a recently published analytic Fourier representation of the level crossing frequency of shot noise processes with piece-wise continuous kernels with jumps. The general analytic result is validated by a Monte Carlo simulation for pulses of the form g(t) = e(-t/τ) (t > 0) and a Gaussian detector response function. The Monte Carlo simulations are in excellent agreement with the analytic predictions of photon counts within the numerical accuracy of the calculations. The phenomenon of pulse pile-up is identified with the level-crossing problem of shot noise processes and an exact, analytic formula for the expected number of counts in energy-sensitive, photon-counting x-ray detectors for arbitrary photon flux, response function, and pulse-shapes is derived. The framework serves as a theoretical foundation for future works on pulse pile-up.

The Level-Crossing Problem: First-Passage, Escape and Extremes

We review the level-crossing problem which includes the first-passage and escape problems as well as the theory of extreme values (the maximum, the minimum, the maximum absolute value and the range or span). We set the definitions and general results and apply them to one-dimensional diffusion processes with explicit results for the Brownian motion and the Ornstein–Uhlenbeck (OU) process.

Extreme values and the level-crossing problem: an application to the Feller process.

We review the question of the extreme values attained by a random process. We relate it to level crossings to one boundary (first-passage problems) as well as to two boundaries (escape problems). The extremes studied are the maximum, the minimum, the maximum absolute value, and the range or span. We specialize in diffusion processes and present detailed results for the Wiener and Feller processes.

Level crossing in a two-photon Jaynes—Cummings model

In this paper, the energy spectrum of the two-photon Jaynes—Cummings model (TPJCM) is calculated exactly in the non-rotating wave approximation (non-RWA), and we study the level-crossing problem by means of fidelity. A narrow peak of the fidelity is observed at the level-crossing point, which does not appear at the avoided-crossing point. Therefore fidelity is perfectly suited for detecting the level-crossing point in the energy spectrum.

Demkov-Kunike model in cold molecule formation: the fast resonance sweep regime of the strong interaction limit

We study a quadratic-nonlinear version of the level-crossing Demkov-Kunike problem relevant to coherent molecule formation via photo- or Feshbach-association of ultracold atoms. Using an exact third-order nonlinear differential equation for the molecular state probability, we construct an approximate solution to the problem in the fast sweep regime of the strong interaction limit. The constructed approximation, defined as a solution of a first-order nonlinear equation, contains a fitting parameter which we determine through a variational procedure. The suggested approach can be interpreted as a modification of the adiabatic approximation and can further be used for the treatment of analogous nonlinear two-state problems.

Landau-Zener problem in a three-level neutrino system with nonlinear time dependence

We consider the level-crossing problem in a three-level system with nonlinearly time-varying Hamiltonian (time-dependence ${t}^{\ensuremath{-}3}$). We study the validity of the so-called independent crossing approximation in the Landau-Zener model by making comparison with results obtained numerically in the density matrix approach. We also demonstrate the failure of the so-called ``nearest zero'' approximation of the Landau-Zener level-crossing probability integral.

A Discrete-Time Approach to Analyze Hard Handoff Performance in Cellular Networks

The handoff algorithm employed in a cellular network has a significant impact on overall network performance, but evaluation of handoff performance has for the most part been done using only crude approximations or brute-force computer simulation. We introduce a new discrete-time approach to analyze the performance of handoff algorithms based on pilot signal strength measurements. We derive exact analytical expressions and develop a recursive numerical procedure to evaluate handoff performance metrics for a mobile station moving along a straight-line trajectory in a cellular network employing hard handoff. The numerical procedure provides a computational solution to a level-crossing problem in discrete time. Our discrete-time approach provides valuable analytical insight into the performance impact of handoff algorithms. Moreover, our numerical procedure for discrete-time handoff analysis provides an accurate and efficient tool for the design and dimensioning of high-performance handoff algorithms. The accuracy of the numerical procedure is validated by simulation.

Competition between Lévy jumps and continuous drift

Motivated by certain birth–death processes with strongly fluctuating birth rates, we consider a level-crossing problem for a random process being a superposition of a continuous drift to the left and jumps to the right. The lengths of the corresponding jumps follow a one-sided extreme Lévy-law of index α. We concentrate on the case 0< α<1 and discuss the probability of crossing a left boundary (“extinction”). We show that this probability decays exponentially as a function of the initial distance to the boundary. Such behavior is universal for all α<1 and is exemplified by an exact solution for the special case α= 1 2 . The splitting probabilities are also discussed.

Functional integration approach to hysteresis.

A general formulation of scalar hysteresis is proposed. This formulation is based on two steps. First, a generating function g(x) is associated with an individual system, and a hysteresis evolution operator is defined by an appropriate envelope construction applied to g(x), inspired by the overdamped dynamics of systems evolving in multistable free-energy landscapes. Second, the average hysteresis response of an ensemble of such systems is expressed as a functional integral over the space G of all admissible generating functions, under the assumption that an appropriate measure mu has been introduced in G. The consequences of the formulation are analyzed in detail in the case where the measure mu is generated by a continuous, Markovian stochastic process. The calculation of the hysteresis properties of the ensemble is reduced to the solution of the level-crossing problem for the stochastic process. In particular, it is shown that, when the process is translationally invariant (homogeneous), the ensuing hysteresis properties can be exactly described by the Preisach model of hysteresis, and the associated Preisach distribution is expressed in closed analytic form in terms of the drift and diffusion parameters of the Markovian process. Possible applications of the formulation are suggested, concerning the interpretation of magnetic hysteresis due to domain wall motion in quenched-in disorder and the interpretation of critical state models of superconducting hysteresis.

Stochastic dynamics in quenched-in disorder and hysteresis

The conditions under which relaxation dynamics in the presence of quenched-in disorder lead to rate-independent hysteresis are discussed. The calculation of average hysteresis branches is reduced to the solution of the level-crossing problem for the stochastic field describing quenched-in disorder. Closed analytical solutions are derived for the case where the disorder is characterized by Wiener–Lévy statistics. This case is shown to be equivalent to the Preisach model and the associated Preisach distribution is explicitly derived, as a function of the parameters describing the original dynamic problem.

Pump-probe spectroscopy of H-bonds: Saturation kinetics and spectral transformation during and after pumping

The saturation kinetics of the H-bond absorption spectra in a strong laser field is described as a level-crossing problem with diffusional motion along the reaction coordinate. The full analogy with kinetics and rate of electron transfer reactions is emphasized when vibrational relaxation to the ground state is negligible. In this case, the transition rate saturates with a field strength reaching the diffusional regime limited by the friction in the ground or excited state. Taking vibrational relaxation into consideration changes qualitatively the kinetics of approaching the stationary level populations and absorption of pumping light. The fast relaxation results in significant desaturation of the light induced transitions and a shortening of the transient time. The absorption spectra of the weak probe light in the context of two-level and three-level models show either loss or gain in the red wing, as well as bleaching and superabsorption in the corresponding transmission spectra. The time evolution of these spectra, after switching the strong field on and off, was studied.

Pump-probe spectroscopy of H-bonds as the level-crossing problem

Saturation of the H-bond absorption spectra by a strong laser field is described as a level-crossing problem with diffusional motion along the reaction coordinate. The stationary populations of the ground and excited vibronic states were found, assuming that the light induced transitions between them occur only at the point of exact resonance and the nonradiative decay of the upper state is uniform. The stationary distributions in both states are also obtained and shown to be Gaussian at slow vibrational relaxation, but essentially distorted when relaxation is faster than diffusion. The absorption spectra of the weak probe light in the frame of two-level and three-level models are calculated, which show either loss or gain in different spectral regions. The existence of bleaching and superabsorption branches in corresponding transmission spectra are confirmed.

Analytic properties and effective two-level problems in stimulated Raman adiabatic passage

We demonstrate that various properties of population transfer by delayed pulses in three-level systems on two-photon resonance can be deduced analytically and for general pulse shapes. We use the fact that the three-level system reduces to effective two-level problems at large intermediate-level detuning \ensuremath{\Delta}, on resonance (\ensuremath{\Delta}=0) and for completely overlapping pulses. Special attention is paid to the effect of the pulse order on the population transfer efficiency. We show that on resonance the transfer efficiency depends substantially on the pulse order, while at large \ensuremath{\Delta} it does not. We also find that under some natural restrictions on the symmetry of the problem, the population of the initial level does not depend on the pulse order at any \ensuremath{\Delta}. Furthermore, we demonstrate that the population transfer in the three-level system can be viewed as a level-crossing problem in an equivalent two-level system not only at large \ensuremath{\Delta} (which is known) but also on resonance, \ensuremath{\Delta}=0. The effective on-resonance two-level problem is interesting by itself as it shows that a level crossing and adiabatic evolution do not necessarily lead to complete population inversion. As examples throughout the paper, we present several analytically solvable models.

Perturbative Approach to Landau-Zener Transitions

The Landau-Zener level-crossing problem is analysed perturbatively. We consider the operator H( t) = Φ( t) σ 3 + Δσ 1, where Φ( t) is an external (time-dependent) force, σ i are the Pauli matrices, and Δ is taken to be a small parameter. By considering the operator U( t, − t) that evolves states in time from − t to t, it is possible to develop a perturbation series in Δ. For three choices of Φ( t), U( t, − t) is determined for small Δ. For Φ( t) periodic with period θ a small Δ approximation to U( t, − t) is obtained that is valid for times of order θ/( θ Δ) 3. A slow modulation of this time evolution operator is found and for certain values of the parameters the slow modulation may stop, leaving U( t, − t) asym 1 for all t. The effects of fluctuations in Φ( t) on the transition probability are also calculated. To O( Δ 2) it is found that for Φ( t) ∝ t, white Gaussian noise modifies the probability of a transition in a finite time but leaves unaltered the long time transition probability. This is in accordance with the findings of Y. Kayanuma ( Phys. Rev. Lett. 58 (1987), 1934).

Dissipation effects on wave packets in level crossings: A comparison between two numerical approaches.

We study the time evolution of wave packets on coupled energy levels including the irreversible effect of spontaneous emission. As a test case we choose a simple two-state level-crossing problem of the Landau-Zener type. The numerical treatment of this situation is demanding, and we explore two approaches based on different views of the quantum-mechanical dynamics. First, we solve the time evolution of the wave packet using the traditional master equation giving directly the ensemble averages. Second, we use the recently developed Monte Carlo simulation method, which gives the individual members of an ensemble of quantum states undergoing spontaneous quantum jumps. The results are compared and found to agree when large enough ensembles are used. We discuss the computational requirements for the two methods and we find both quite demanding but qualitatively different.

Zero-crossing rates of functions of Gaussian processes

Formulas for the expected zero-crossing rates of random processes that are monotone transformations of Gaussian processes can be obtained by using two different techniques. The first technique involves derivation of the expected zero-crossing rate for discrete-time processes and extends the result of the continuous-time case by using an appropriate limiting argument. The second is a direct method that makes successive use of R. Price's (1958) theorem, the chain rule for derivatives, and S.O. Rice's (1954) formula for the expected zero-crossing rate of a Gaussian process. A constant, which depends on the variance of the transformed process and a second-moment of its derivative, is derived. Multiplying Rice's original expression by this constant yields the zero-crossing formula for the transformed process. The two methods can be used for the general level-crossing problem of random processes that are monotone functions of a Gaussian process. >

Adiabatic conversion of solar neutrinos.

The adiabatic conversion of solar electron neutrinos to muon neutrinos via the Mikheyev-Smirnov-Wolfenstein mechanism is discussed as a level-crossing problem. Solutions to the /sup 37/Cl puzzle obtained numerically by Mikheyev and Smirnov, Rosen and Gelb, and others are shown to form the boundary of the region in the deltam/sup 2/-sin/sup 2/2theta plane where adiabatic conversion occurs. Most of the region within this boundary is ruled out by the nonzero signal in Davis's experiment. The Landau-Zener description of the adiabatic boundary is employed.

Aperture-averaged spectral correlations of beams in a turbulent atmosphere

The influence of aperture averaging on the two-frequency correlation of a beam that propagates in a weakly turbulent atmosphere is considered. A computational scheme to derive closed expressions for the aperture-averaged two-frequency log amplitude and phase correlation functions is presented. An application of the resulting expressions to a level-crossing problem is illustrated.

A note on first passage time problems for Gaussian processes and varying boundaries (Corresp.)

First and second-order approximations to the first passage time conditional probability density function of a stationary Gaussian process, with differentiable sample paths crossing a time-dependent boundary, are explicitly provided. In the limit of infinitely large time, our results coincide with the well-known results of Kac and Rice for the constant level-crossing problem.