arXiv subjects

Saar Rahav

Publications and source records attributed to Saar Rahav.

44 records · Page 3Linked to original sources

arXiv2005

Time-independent approximations for periodically driven systems with friction

The classical dynamics of a particle that is driven by a rapidly oscillating potential (with frequency $ω$) is studied. The motion is separated into a slow part and a fast part that oscillates around the slow part. The motion of the slow part is found to be described by a time-independent equation that is derived as an expansion in orders of $ω^{-1}$ (in this paper terms to the order $ω^{-3}$ are calculated explicitly). This time-independent equation is used to calculate the attracting fixed points and their basins of attraction. The results are found to be in excellent agreement with numerical solutions of the original time-dependent problem.

nlin.CD↗

arXiv2003

Point perturbations of circle billiards

The spectral statistics of the circular billiard with a point-scatterer is investigated. In the semiclassical limit, the spectrum is demonstrated to be composed of two uncorrelated level sequences. The first corresponds to states for which the scatterer is located in the classically forbidden region and its energy levels are not affected by the scatterer in the semiclassical limit while the second sequence contains the levels which are affected by the point-scatterer. The nearest neighbor spacing distribution which results from the superposition of these sequences is calculated analytically within some approximation and good agreement with the distribution that was computed numerically is found.

nlin.CD↗

arXiv2003

Time independent description of rapidly oscillating potentials

The classical and quantum dynamics in a high frequency field are found to be described by an effective time independent Hamiltonian. It is calculated in a systematic expansion in the inverse of the frequency ($ω$) to order $ω^{-4}$. The work is an extension of the classical result for the Kapitza pendulum, which was calculated in the past to order $ω^{-2}$. The analysis makes use of an implementation of the method of separation of time scales and of a quantum gauge transformation in the framework of Floquet theory. The effective time independent Hamiltonian enables one to explore the dynamics in presence of rapidly oscillating fields, in the framework of theories that were developed for systems with time independent Hamiltonians. The results are relevant, in particular, for exploration of the dynamics of cold atoms.

nlin.CD↗

arXiv2003

Effective Hamiltonians for periodically driven systems

The dynamics of classical and quantum systems which are driven by a high frequency ($ω$) field is investigated. For classical systems the motion is separated into a slow part and a fast part. The motion for the slow part is computed perturbatively in powers of $ω^{-1}$ to order $ω^{-4}$ and the corresponding time independent Hamiltonian is calculated. Such an effective Hamiltonian for the corresponding quantum problem is computed to order $ω^{-4}$ in a high frequency expansion. Its spectrum is the quasienergy spectrum of the time dependent quantum system. The classical limit of this effective Hamiltonian is the classical effective time independent Hamiltonian. It is demonstrated that this effective Hamiltonian gives the exact quasienergies and quasienergy states of some simple examples as well as the lowest resonance of a non trivial model for an atom trap. The theory that is developed in the paper is useful for the analysis of atomic motion in atom traps of various shapes.

nlin.CD↗

arXiv2002

Spectral Statistics of Rectangular Billiards with Localized Perturbations

The form factor $K(τ)$ is calculated analytically to the order $τ^3$ as well as numerically for a rectangular billiard perturbed by a $δ$-like scatterer with an angle independent diffraction constant, $D$. The cases where the scatterer is at the center and at a typical position in the billiard are studied. The analytical calculations are performed in the semiclassical approximation combined with the geometrical theory of diffraction. Non diagonal contributions are crucial and are therefore taken into account. The numerical calculations are performed for a self adjoint extension of a $δ$ function potential. We calculate the angle dependent diffraction constant for an arbitrary perturbing potential $U({\bf r})$, that is large in a finite but small region (compared to the wavelength of the particles that in turn is small compared to the size of the billiard). The relation to the idealized model of the $δ$-like scatterer is formulated. The angle dependent diffraction constant is used for the analytic calculation of the form factor to the order $τ^2$. If the scatterer is at a typical position, the form factor is found to reduce (in this order) to the one found for angle independent diffraction. If the scatterer is at the center, the large degeneracy in the lengths of the orbits involved leads to an additional small contribution to the form factor, resulting of the angle dependence of the diffraction constant. The robustness of the results is discussed.

nlin.CD↗

arXiv2002

Relaxation and Noise in Chaotic Systems

For a class of idealized chaotic systems (hyperbolic systems) correlations decay exponentially in time. This result is asymptotic and rigorous. The decay rate is related to the Ruelle-Pollicott resonances. Nearly all chaotic model systems, that are studied by physicists, are not hyperbolic. For many such systems it is known that exponential decay takes place for a long time. It may not be asymptotic, but it may persist for a very long time, longer than any time of experimental relevance. In this review a heuristic method for calculation of this exponential decay of correlations in time is presented. It can be applied to model systems, where there are no rigorous results concerning this exponential decay. It was tested for several realistic systems (kicked rotor and kicked top) in addition to idealized systems (baker map and perturbed cat map). The method consists of truncation of the evolution operator (Frobenius-Perron operator), and performing all calculations with the resulting finite dimensional matrix. This finite dimensional approximation can be considered as coarse graining, and is equivalent to the effect of noise. The exponential decay rate of the chaotic system is obtained when the dimensionality of the approximate evolution operator is taken to infinity, resulting in infinitely fine resolution, that is equivalent to vanishing noise. The corresponding Ruelle-Pollicott resonances can be calculated for many systems that are beyond the validity of the Ruelle-Pollicott theorem.

nlin.CD↗

arXiv2001

Localized Perturbations of Integrable Systems

The statistics of energy levels of a rectangular billiard, that is perturbed by a strong localized potential, are studied analytically and numerically, when this perturbation is at the center or at a typical position. Different results are found for these two types of positions. If the scatterer is at the center, the symmetry leads to additional contributions, some of them are related to the angular dependence of the potential. The limit of the $δ$-like scatterer is obtained explicitly. The form factor, that is the Fourier transform of the energy-energy correlation function, is calculated analytically, in the framework of the semiclassical geometrical theory of diffraction, and numerically. Contributions of classical orbits that are non diagonal are calculated and are found to be essential.

nlin.CD↗

arXiv1999

The Accuracy of Semiclassical Quantization for Integrable Systems

The eigenvalues of the Hyperspherical billiard are calculated in the semiclassical approximation. The eigenvalues where this approximation fails are identified and found to be related to caustics that approach the wall of the billiard. The fraction of energy levels for which the semiclassical error is larger than some given value is calculated analytically (and tested numerically) and found to be independent of energy. The implications for other systems, in particular integrable ones, are discussed.

chao-dyn↗