Homoclinic signatures of dynamical localization

Abstract

It is demonstrated that the oscillations in the width of the momentum distribution of atoms moving in a phase-modulated standing light field, as a function of the modulation amplitude λ, are correlated with the variation of the chaotic layer width in energy of an underlying effective pendulum. The maximum effect of dynamical localization and the nearly perfect delocalization are associated with the maxima and minima, respectively, of the chaotic layer width. It is also demonstrated that kinetic energy is conserved as an almost adiabatic invariant at the minima of the chaotic layer width, and that the system is accurately described by delta-kicked rotors at sufficiently large zeros of the Bessel functions J 0 (λ) and J 1 (λ). Numerical calculations of kinetic energy and Lyapunov exponents confirm all the theoretical predictions.PACS numbers: 05.45. Mt, 42.50.Vk Nonautonomous Hamiltonian systems exhibit a wide range of novel effects with different manifestations in the quantum and classical domains [1,2]. A situation of especial interest occurs when the classical dynamics undergoes a transition from stability to chaos as a control parameter is varied [3]. In this regard, dynamical localization (DL) is a specifically quantum phenomenon taking place in systems where the classical dynamics exhibits chaotic diffusion in momentum space, while the asymptotic quantum momentum distribution freezes to a steady state although it is initially similar to its classical counterpart. This phenomenon is attributed to quantum interferences among the diffusive paths which on average over long times are completely destructive. These interferences giving rise to chaotic diffusion are properly described in terms of gradual dephasing of the various Floquet states that contribute to the initial state. After a characteristic time (the so-called Ehrenfest time

Towards AC-induced optimum control of dynamical localization

“…The classical momentum distribution for our system is numerically calculated as where P (x, p; τ ) is the evolution of a uniform distribution over one wavelength with a Gaussian momentum distribution characterized by a width of ∆p 0 = 0.386 [4][5][6] given by the corresponding Liouville equation, defined as…”

“…Quantum effects in transport phenomena in classical systems represent an interesting fundamental issue in quantum theory that started from a remark of Einstein in his celebrated paper on torus quantization [1]. One of such effects, widely studied in the context of timeperiodic systems, is the quantum suppression of classical chaotic diffusion [2][3][4][5][6][7], or dynamical localization (DL) for short. Remarkably, this effect is a quantum manifestation of the fact that a time-periodic force can stabilize a system, and it is thus expected to play a key role in our understanding of the problem of quantum-classical correspondence in classically chaotic systems [8,9].…”

“…( 13) below) flexible periodic pulse which varies its form depending on the value of m. For m = 0, one recovers the well-known harmonic excitation case previously considered for example in refs. [5][6][7], since then F (t; m = 0, T ) = sin (2πt/T ). On the other hand, for m = 0 the waveform has different shapes.…”

Universal chaotic layer width in space-periodic Hamiltonian systems under adiabatic ac time-periodic forces

“…It has also been shown that the width of the SCL is related to the average current [11], and that (a) E-mail: rchacon@unex.es adiabatically driven Hamiltonian ratchets allow for giant acceleration of particles [12][13][14][15] because of the divergence of the chaotic layer. It has also been demonstrated that the strength of the dynamic localization of atoms moving in a phase-modulated standing light field is correlated with the chaotic layer width of a perturbed pendulum that effectively describes the atoms' dynamics [16]. Notably, such a correlation was found to hold irrespective of the modulation's waveform [17].…”

“…Remarkably, this provides an additional degree of freedom to maximally optimize the control of the SCL width and the strength of the related chaotic spatial transport -via facilitation of global chaos-with possible applications to noise-induced transport phenomena (see [25] and references therein) for example. Since there are no significant islands of stability within the SCL, we can assume that chaotic diffusion is nearly ergodic and the present theory can be thus directly implemented to the problem of quantum transport in the semiclassical regime, relying upon the theorems of Schnirelman, Colin de Verdiere and Zelditch [26][27][28] and the relationship between the strength of dynamical localization and the chaotic layer width [16,17]. Also, eq.…”