arXiv subjects

Itzhack Dana

Publications and source records attributed to Itzhack Dana.

27 records · Page 2Linked to original sources

arXiv2006

Fluctuations and Transients in Quantum-Resonant Evolution

The quantum-resonant evolution of the mean kinetic energy (MKE) of the kicked particle is studied in detail on different time scales for {\em general} kicking potentials. It is shown that the asymptotic time behavior of a wave-packet MKE is typically a linear growth with bounded fluctuations having a simple number-theoretical origin. For a large class of wave packets, the MKE is shown to be exactly the superposition of its asymptotic behavior and transient logarithmic corrections. Both fluctuations and transients can be significant for not too large times but they may vanish identically under some conditions. In the case of incoherent mixtures of plane waves, it is shown that the MKE never exhibits asymptotic fluctuations but transients usually occur.

nlin.CD↗

arXiv2005

General Approach to the Quantum Kicked Particle in a Magnetic Field: Quantum-Antiresonance Transition

The quantum kicked particle in a magnetic field is studied in a weak-chaos regime under realistic conditions, i.e., for {\em general} values of the conserved coordinate $x_{\rm c}$ of the cyclotron orbit center. The system exhibits spectral structures [``Hofstadter butterflies'' (HBs)] and quantum diffusion depending sensitively on $x_{\rm c}$. Most significant changes take place when $x_{\rm c}$ approaches the value at which quantum antiresonance (exactly periodic recurrences) can occur: the HB essentially ``doubles'' and the quantum-diffusion coefficient $D(x_{\rm c})$ is strongly reduced. An explanation of these phenomena, including an approximate formula for $D(x_{\rm c})$ in a class of wave packets, is given on the basis of an effective Hamiltonian which is derived as a power expansion in a small parameter. The global quantum diffusion of a two-dimensional wave packet for all $x_{\rm c}$ is briefly considered.

nlin.CD↗

arXiv2005

General Quantum Resonances of the Kicked Particle

The quantum resonances (QRs) of the kicked particle are studied in a most general framework by also considering {\em arbitrary} periodic kicking potentials. It is shown that QR can arise, in general, for {\em any rational} value of the Bloch quasimomentum. This is illustrated in the case of the main QRs for arbitrary potentials. In this case, which is shown to be precisely described by the linear kicked rotor, exact formulas are derived for the diffusion coefficients determining the asymptotic evolution of the average kinetic energy of either an incoherent mixture of plane waves or a general wave packet. The momentum probability distribution is exactly calculated and studied for a two-harmonic potential. It clearly exhibits new resonant values of the quasimomentum and it is robust under small deviations from QR.

nlin.CD↗

arXiv2003

Global Superdiffusion of Weak Chaos

A class of kicked rotors is introduced, exhibiting accelerator-mode islands (AIs) and {\em global} superdiffusion for {\em arbitrarily weak} chaos. The corresponding standard maps are shown to be exactly related to generalized web maps taken modulo an ``oblique cylinder''. Then, in a case that the web-map orbit structure is periodic in the phase plane, the AIs are essentially {\em normal} web islands folded back into the cylinder. As a consequence, chaotic orbits sticking around the AI boundary are accelerated {\em only} when they traverse tiny {\em ``acceleration spots''}. This leads to chaotic flights having a quasiregular {\em steplike} structure. The global weak-chaos superdiffusion is thus basically different in nature from the strong-chaos one in the usual standard and web maps.

nlin.CD↗

arXiv2003

Chaotic Diffusion on Periodic Orbits: The Perturbed Arnol'd Cat Map

Chaotic diffusion on periodic orbits (POs) is studied for the perturbed Arnol'd cat map on a cylinder, in a range of perturbation parameters corresponding to an extended structural-stability regime of the system on the torus. The diffusion coefficient is calculated using the following PO formulas: (a) The curvature expansion of the Ruelle zeta function. (b) The average of the PO winding-number squared, $w^{2}$, weighted by a stability factor. (c) The uniform (nonweighted) average of $w^{2}$. The results from formulas (a) and (b) agree very well with those obtained by standard methods, for all the perturbation parameters considered. Formula (c) gives reasonably accurate results for sufficiently small parameters corresponding also to cases of a considerably nonuniform hyperbolicity. This is due to {\em uniformity sum rules} satisfied by the PO Lyapunov eigenvalues at {\em fixed} $w$. These sum rules follow from general arguments and are supported by much numerical evidence.

nlin.CD↗

arXiv2000

Renormalization of Quantum Anosov Maps: Reduction to Fixed Boundary Conditions

A renormalization scheme is introduced to study quantum Anosov maps (QAMs) on a torus for general boundary conditions (BCs), whose number ($k$) is always finite. It is shown that the quasienergy eigenvalue problem of a QAM for {\em all} $k$ BCs is exactly equivalent to that of the renormalized QAM (with Planck's constant $\hbar ^{\prime}=\hbar /k$) at some {\em fixed} BCs that can be of four types. The quantum cat maps are, up to time reversal, fixed points of the renormalization transformation. Several results at fixed BCs, in particular the existence of a complete basis of ``crystalline'' eigenstates in a classical limit, can then be derived and understood in a simple and transparent way in the general-BCs framework.

nlin.CD↗

arXiv1999

Exact Renormalization Scheme for Quantum Anosov Maps

An exact renormalization scheme is introduced for quantum Anosov maps (QAMs) on a torus for general boundary conditions (BCs), whose number is always finite. Given a QAM $\hat{U}$ with $k$ BCs and Planck's constant $\hbar =2π/p$ ($p$ integer), its $n$th renormalization iterate $\hat{U}^{(n)}={\cal R}^{n}(\hat{U})$ is associated with $k$ BCs for all $n$ and with a Planck's constant $\hbar ^{(n)}=\hbar /k^{n}$. It is shown that the quasienergy eigenvalue problem for $\hat{U}^{(n)}$ for {\em all} $k$ BCs is equivalent to that for $\hat{U}^{(n+1)}$ at some {\em fixed} BCs, corresponding, for $n>0$, to either strict {\em periodicity} for $kp$ even or {\em antiperiodicity} for $kp$ odd. The quantum cat maps are, in general, fixed points of either ${\cal R}$ or ${\cal R}^{2}$. The Hannay-Berry results turn out then to be significant also for general BCs.

chao-dyn↗

arXiv1998

Band Distributions for Quantum Chaos on the Torus

Band distributions (BDs) are introduced describing quantization in a toral phase space. A BD is the uniform average of an eigenstate phase-space probability distribution over a band of toral boundary conditions. A general explicit expression for the Wigner BD is obtained. It is shown that the Wigner functions for {\em all} of the band eigenstates can be reproduced from the Wigner BD. Also, BDs are shown to be closer to classical distributions than eigenstate distributions. Generalized BDs, associated with sets of adjacent bands, are used to extend in a natural way the Chern-index characterization of the classical-quantum correspondence on the torus to arbitrary rational values of the scaled Planck constant.

chao-dyn↗

arXiv1998

Band Husimi Distributions and the Classical-Quantum Correspondence on the Torus

Band Husimi distributions (BHDs) are introduced in the quantum-chaos problem on a toral phase space. In the framework of this phase space, a quantum state must satisfy Bloch boundary conditions (BCs) on a torus and the spectrum consists of a finite number of levels for given BCs. As the BCs are varied, a level broadens into a band. The BHD for a band is defined as the uniform average of the Husimi distributions for all the eigenstates in the band. The generalized BHD for a set of adjacent bands is the average of the BHDs associated with these bands. BHDs are shown to be closer, in several aspects, to classical distributions than Husimi distributions for individual eigenstates. The generalized BHD for two adjacent bands is shown to be approximately conserved in the passage through a degeneracy between the bands as a nonintegrability parameter is varied. Finally, it is shown how generalized BHDs can be defined so as to achieve physical continuity under small variations of the scaled Planck constant. A generalization of the topological (Chern-index) characterization of the classical-quantum correspondence is then obtained.

chao-dyn↗