In this work we consider the effects of noise on system behavior during parameter sweeps through bifurcations that
induce sharp jumps in response amplitudes. These problems arise in a variety of applications, including the use of
bifurcation amplification in micro-sensors. Due to inherent system noise, the observed bifurcation events are stochastic
in nature and one must estimate parameter values from a distribution. A stochastic dynamical systems analysis allows
one to distill the problem to a one-dimensional, one-parameter Fokker-Planck equation, where the parameter is the ratio
of the noise intensity to sweep rate. Approximate closed form solutions for the distributions are obtained in the limits of
slow and fast parameter sweep rates, and a numeric solution captures the intermediate sweep range that bridges these two
approximations. These results are essential for quantifying errors in bifurcation amplifiers and for optimizing bifurcation
detection schemes, as used in sensing applications. Preliminary experimental results for a parametrically excited microdevice
show good qualitative agreement with the theory.
KEYWORDS: Modulation, Particles, Diffusion, Monte Carlo methods, Solids, Systems modeling, Control systems, Stochastic processes, Numerical analysis, Motion models
We provide a complete solution of the problem of noise-induced escape in periodically driven systems. We show that both the exponent and the prefactor in the escape rate display scaling behavior with the field intensity. The corresponding scaling is related to synchronization of escape events by the modulating field. The onset of the synchronization with the increasing field and its loss as the field approaches a bifurcational value lead to a strongly nonmonotonic field dependence of the prefactor.
We analyze localization of interacting excitations in a system of qubits or spins. The system is modeled by a spin chain with an anisotropic (XXZ) exchange coupling in a magnetic field. Localization occurs on a defect with an excess on-site spin-flip energy. Such a defect corresponds to a qubit with the level spacing different from other qubits. Because of the interaction, a single defect may lead to multiple localized states. We find energy spectra and localization lengths of the two-excitation states. An excitation remains localized on the defect even where energy conservation allows scattering into extended states. This is due to destructive quantum interference in the two-excitation scattering channels, and it facilitates the operation of a quantum computer. Analytical results are obtained for strong anisotropy and are confirmed by numerical studies.
Strong many-particle localization is studied in a 1D array of perpetually coupled qubits and an equivalent 1D system of interacting fermions. We construct a bounded sequence of the on-site fermion energies, or qubit transition frequencies, that suppresses resonant hopping between both nearest and remote neighbors. Besides quasi-exponential decay of the single-particle transition amplitude,it leads to long lived strongly localized many-particle states. This makes quantum computing with time-independent qubit coupling viable.
An application of the path-integral approach to an analysis of the
fluctuations in complex dynamical systems is discussed. It is
shown that essentially the same ideas underly recent progress in
the solutions of a number of long-standing problems in complex
dynamics. In particular, we consider the problems of prediction,
control and inference of chaotic dynamics perturbed by noise in
the framework of path-integral approach.
We study the rate of activated escape W in periodically modulated systems close to the saddle-node bifurcation point where the metastable state disappears. The escape rate displays scaling behavior versus modulation amplitude A as A approaches the bifurcational value Ac, with 1nW ∝(Ac-A)μ. For adiabatic modulation, the critical exponent is μ=3/2. Even if the modulation is slow far from the bifurcation point, the adiabatic approximation breaks down close to Ac. In the weakly nonadiabatic regime we predict a crossover to μ = 2 scaling. For higher driving frequencies, as Ac is approached there occurs another crossover, from Αμ=2 to μ=3/2. The general results are illustrated using a simple model system.
Access to the requested content is limited to institutions that have purchased or subscribe to SPIE eBooks.
You are receiving this notice because your organization may not have SPIE eBooks access.*
*Shibboleth/Open Athens users─please
sign in
to access your institution's subscriptions.
To obtain this item, you may purchase the complete book in print or electronic format on
SPIE.org.
INSTITUTIONAL Select your institution to access the SPIE Digital Library.
PERSONAL Sign in with your SPIE account to access your personal subscriptions or to use specific features such as save to my library, sign up for alerts, save searches, etc.