Critical phenomena of dynamical delocalization in quantum maps: Standard map and Anderson map

Abstract

Following the paper exploring the Anderson localization of monochromatically perturbed kicked quantum maps [Phys.Rev. E97,012210], the delocalization-localization transition phenomena in polychromatically perturbed quantum maps (QM) is investigated focusing particularly on the dependency of critical phenomena upon the number M of the harmonic perturbations, where M +1 = d corresponds to the spatial dimension of the ordinary disordered lattice. The standard map and the Anderson map are treated and compared. As the basis of analysis, we apply the self-consistent theory (SCT) of the localization, taking a plausible hypothesis on the mean-free-path parameter which worked successfully in the analyses of the monochromatically perturbed QMs. We compare in detail the numerical results with the predictions of the SCT, by largely increasing M . The numerically obtained index of critical subdiffusion t α (t:time) agrees well with the prediction of one-parameter scaling theory α = 2/(M + 1), but the numerically obtained critical exponent of localization length significantly deviates from the SCT prediction. Deviation from the SCT prediction is drastic for the critical perturbation strength of the transition: if M is fixed the SCT presents plausible prediction for the parameter dependence of the critical value, but its value is 1/(M − 1)times smaller than the SCT prediction, which implies existence of a strong cooperativity of the harmonic perturbations with the main mode.

Localization and delocalization properties in quasi-periodically perturbed Kicked Harper and Harper models

“…It also provides insight into the control of localized and delocalized states by coherent perturbations in the Floquet engineering. Loc Loc Loc LDT LDT Kicked Anderson model [38] Loc Loc LDT LDT LDT KIcked rotor [38] Loc Balli Balli BDT? BDT BDT Kicke Anderson model [39] Balli Loc Diff Diff Diff Kicked Harper model (V >> 1) Balli BDT BDT BDT BDT…”

“…It is also feasible to experimentally realize the dynamical localization transition through the diffusion of wave packets in the pulsed 1D incommensurate optical lattice such as the KHM. [33] On the other hand, we also investigated the characteristics of the LDT in the polychromatically perturbed kicked Anderson model (KAM), and the results that correspond well with those in the KR have been obtained [34][35][36][37][38][39]. Note that in our previous paper, we used the expression Anderson map (AM) for the kicked Anderson model (KAM).…”

“…γ and |u are the quasieigenvalue and quasi-eigenstate. Here, if the eigenstate representation of Ĵj is used, Ĵj |m j = m j |m j (m j ∈ Z), we can obtain the following (M + 1)−dimensional tightbinding expression by the Maryland transform [38]:…”

Localized-Diffusive and Ballistic-Diffusive Transitions in Kicked Incommensurate lattice

“…Introduction.-In 1D tight-binding model with any disorder the dynamical quantum states are always localized [1] But if it is coupled with noisy source such as irregular thermal lattice vibrations, the localization is immediately destroyed and is taken the place by normal diffusion [2,3]. Even very simple time-dependent harmonic perturbations consisting only a few number of periods may convert the localized wave packet into irreversible diffusion in Anderson model and kicked Anderson model (Anderson map) [4][5][6]. The characteristic feature of such coherent perturbations is that it realizes irreversible diffusion from the localized state via a critical phase transition we call localization-delocalization transition (LDT).…”

“…The purpose of this paper is to show realization of the normal diffusion by simple harmonic perturbations via the above two transitions paths, where the harmonic perturbation is polychromatic and is composed of M frequencies mode. We first show the localized state of KHM responds to such perturbation in quite similar ways to that of kicked disordered lattice systems such as the Anderson map [4,5] and quantum standard map [5,[12][13][14]: for monochromatic perturbation M = 1 the localizaion still remains and it is M ≥ 2 that the KHM localized state undergoes the LDT with increase in the perturbation strength ǫ. We next show the response of the extended (ballistic) state of KHM to the harmonic perturbation.…”