Universality of the Anderson Transition with the Quasiperiodic Kicked Rotor

We report a numerical analysis of the Anderson transition in a quantum-chaotic system, the quasiperiodic kicked rotor with three incommensurate frequencies. It is shown that this dynamical system exhibits the same critical phenomena as the truly random 3D-Anderson model. By taking proper account of systematic corrections to one-parameter scaling, the universality of the critical exponent is demonstrated. Our result ν = 1.59± 0.01 is in perfect agreement with the value found for the Anderson model. Introduction. – It is now widely acknowledged that the classical diffusive behavior of non-interacting electrons in a disordered potential can be stopped by non-trivial interference effects [1]. This puzzling phenomenon, Anderson localization, constitutes one strong evidence of the very difference between quantum and classical dynamics of complex systems. In the field of quantum chaos, a similar phenomenon is acknowledged, namely dynamical localization. In the kicked rotor, a paradigmatic system of quantum chaos, quantum mechanical interference tend to suppress the classical chaotic diffusive dynamics. The discovery of the parallel between dynamical localization and Anderson localization originated from the mapping of the kicked rotor to the quasirandom 1D Anderson model [2]. In Ref. [3] it was demonstrated that the kicked rotor in the dynamical localization regime could be modeled by random band matrices; the latter have been reduced to a 1D nonlinear σ model [4] similar to those employed in the localization theory [5]. In Ref. [6] the direct correspondence between the kicked rotor and the diffusive supersymmetric nonlinear σ model was demonstrated. In the localized regime, the kicked rotor exactly mimics the behavior of disordered electronic conductors. There has been much efforts to observe Anderson localization in 3D experimentally. However, due to stray effects like interaction, decoherence or absorption, very few attempts have been successful [7]. In a slightly different context, Anderson localization of acoustic [8] and electromagnetic [9–12] waves has been experimentally observed. The experimental realization of the kicked rotor with lasercooled atoms interacting with a pulsed standing wave allowed for the first experimental observation of Anderson localization in 1D with atomic matter waves [13]. One step further is to observe the well-known Anderson transition with this type of system, i.e. the disorder induced metalinsulator transition predicted for non-interacting electrons subjected to a 3D disordered potential. Different generalizations of the kicked rotor have been theoretically considered as analogs of the 3D-Anderson model [14]. Here, we focus on the convenient three-incommensurate-frequencies generalization introduced in Ref. [15]. Very recently an experiment based on this system has fully characterized the Anderson metal-insulator transition [16]. Indeed, a careful analysis of the scaling properties of the dynamics has resulted in the first experimental determination of the localization length critical exponent ν of the Anderson transition. The value found ν = 1.4 ± 0.3 is compatible with the precedent numerical determination of ν = 1.57± 0.02 for the true-random 3D-Anderson model [17]. At this stage, the equivalence between the quasiperiodic kicked rotor [16] and 3D-disordered conductors still has the status of a conjecture (see [18]). A rigorous answer to the question whether this dynamical system exhibits the same critical phenomena – i.e. belongs to the