Itzhack Dana

Experimental Realization of Quantum-Resonance Ratchets at Arbitrary Quasimomenta

Quantum-resonance ratchets associated with the kicked particle are experimentally realized for arbitrary quasimomentum using a Bose-Einstein condensate (BEC) exposed to a pulsed standing light wave. The ratchet effect for general quasimomentum arises even though both the standing-wave potential and the initial state of the BEC have a point symmetry. The experimental results agree well with theoretical ones which take into account the finite quasimomentum width of the BEC. In particular, this width is shown to cause a suppression of the ratchet acceleration for exactly resonant quasimomentum, leading to a saturation of the directed current.

Quantised Hall conductance in a perfect crystal

Using magnetic translation symmetry, the Hall conductance of an isolated magnetic band in units of e2/h is shown to satisfy the Diophantine equation p sigma +qm=1, where p and q are relatively prime integers giving the number of flux quanta per unit cell area, phi =p/q, and m is an integer. This equation holds for a general periodic Schrodinger Hamiltonian with an arbitrary magnetic field and is a direct consequence of the q-fold degeneracy of magnetic bands. Extension to general real phi gives the equation phi sigma H- rho =integer with sigma H the Hall conductance and rho the number of electrons per unit cell, from which sigma H is uniquely determined once rho , phi and the gap structure are given.

Resonances and transport in the sawtooth map

We study transport in a completely chaotic Hamiltonian system, the hyperbolic sawtooth map. Analytical expressions are obtained for its cantori and resonances. We show that resonances give a complete partition of phase space. The flux leaking out of a resonance is given by its turnstiles, whose form and areas are obtained analytically. When the total flux out of a resonance becomes one third the area of an island, the topology of the turnstiles changes. At the same parameter value, a horseshoe is formed corresponding to the orbits trapped within the resonance. Based on this, a coding scheme for the trapped orbits is introduced and expressions for trapped ordered orbits are obtained. The partial flux transferred from one resonance to another is determined by the degree of overlap of their turnstiles. We calculate the survival probability within a resonance using the Markov model; the results are compared with results obtained numerically and from periodic-orbit theory.

Hamiltonian transport on unstable periodic orbits

Transport in Hamiltonian systems is approached on the basis of the unstable periodic orbits (UPOs) embedded in the chaotic region. Correlation functions and diffusion rates are defined for UPO ensembles of two-dimensional periodic maps. Global diffusion in these maps is interpreted in terms of the ensembles UT, containing all the UPOs of period T: as T → ∞, the relative populations of closed orbits and accelerator modes in UT approach a Gaussian distribution, characterized by the diffusion coefficient D. Evidence for the validity of this mechanism is provided for the cat and sawtooth maps. Prime-lattice ensembles of the cat maps appear to be optimal in reproducing the global diffusion. These ensembles are, generically, small subsets of UT for some T. It is shown, however, that the corresponding diffusion rates assume, for T odd, precisely the known value of D in the entire time interval t ≤ T. For T even, all orbits are closed, and D is reproduced only in the interval t ≤ 12 T. The diffusion rates associated with ensembles UT of the sawtooth maps appear to approximate well the value of D if T is of the order of the diffusion time. The case of local transport rates, associated with ensembles that are localized in phase space, is illustrated.

Diffusion in the standard map

Abstract Diffusion in the standard map is studied numerically. The stochasticity parameter K is near the critical value K c , and the diffusion coefficient D is calculated. It is found to satisfy to a good approximation the scaling relation D ∝ ( K − K c ) η , with η in good agreement with the value predicted by the scaling theory of the disappearance of the last bounding KAM torus. The critical region where this scaling relation holds is surprisingly large, i.e. K ≤ 2.5. The mechanism of transport from the chaotic region to the remnants of the last KAM torus is investigated. Evidence for the existence of a narrow stochastic channel mediating this transport is presented and its origin is discussed. Although the scaling of D agrees with the predictions of the scaling theory the transport mechanism is different from the one assumed in this theory.

Vortex-lattice wave fields

Abstract Vortex-lattice wave fields (VLWs), with arbitrary vorticity N per unit cell, are introduced into Fourier optics. Expressions for nondiffracting VLWs, with pre-determined vortex positions in the unit cell, are given. Provided a special phase factor is associated with the VLW, the corresponding source distribution turns out to be a quasiperiodic function of vorticity N or − N . A general solution to the problem of self-Fourier VLWs, in particular, rotationally symmetric VLWs, is obtained. As a result of the rapid phase oscillations of a VLW, its autocorrelation function is usually a 2D Dirac comb modulated by a decaying envelope function.

Adiabatic invariance in the standard map

Adiabatic invariance of the action is investigated in the standard map, under slow changes of the stochasticity parameter K. A fixed action representation of the rotational tori is developed perturbatively in K, and is connected with the usual (KAM) representation at fixed winding number. The notion of adiabatic invariance to a given order in K, K⪡1, is then introduced. It involves an approximation to exact dynamics by essentially a power-series expansion in both K and a slowness parameter. Adiabatic invariance is explicitly verified to second order, and the dependence of the nonadiabaticity on the form of the switching function and the slowness of the change is investigated. The case of adiabatic switching to larger values of K, K<1, is approached phenomenologically, taking into account the problem of separatrix crossing. It is shown that this crossing leads in general to nonadiabatic effects in the limit of infinitely slow change. These reflect the finite widths, or intrinsic action uncertainties, associated with the main island chains crossed. The latter are determined from the Farey tree in the range of variation of the winding number. An estimate of the critical slowness parameter, corresponding to the onset of the intrinsic nonadiabatic effects, is derived. It involves typical time-scales associated with the main island chains crossed.

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 arbitrary periodic kicking potentials. It is shown that QR can arise, in general, for 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 additional resonant values of the quasimomentum and it is robust under small deviations from QR.

Kicked Harper models and kicked charge in a magnetic field

Abstract The symmetric kicked Harper (KH) model was orginally derived as a first-order approximation (in the kicking parameter) of the problem of a kicked charge in a magnetic field (KCM). It is shown that all generalized KH models (symmetric or non-symmetric, periodic or not) can, in fact, be exactly related to KCM problems. Because of this exact relation, all the results derived recently for KH models hold straightforwardly also for the corresponding KCM problems. For the vast majority of KCM problems, however, there exist no generalized KH models related to them.

Topologically universal spectral hierarchies of quasiperiodic systems

Topological properties of energy spectra of general one-dimensional quasiperiodic systems, describing also Bloch electrons in magnetic fields, are studied for an infinity of irrational modulation frequencies corresponding to irrational numbers of flux quanta per unit cell. These frequencies include well-known ones considered in works on Fibonacci quasicrystals. It is shown that the spectrum for any such frequency exhibits a self-similar hierarchy of clusters characterized by universal (system-independent) values of Chern topological integers which are exactly determined. The cluster hierarchy provides a simple and systematic organization of all the spectral gaps, labeled by universal topological numbers which are exactly determinable, thus avoiding their numerical evaluation using rational approximants of the irrational frequency. These numbers give both the quantum Hall conductance of the system and the winding number of the edge-state energy traversing a gap as a Bloch quasimomentum is varied.