We investigate the quantum walk and the quantum kicked rotor in resonance subjected to noise with a Lévy waiting time distribution. We find that both systems have a sub-ballistic wave function spreading as shown by a power-law tail of the standard deviation.PACS numbers: PACS: 03.67. 05.40.Fb; 05.45.Mt In the last decades the study of simple quantum systems, such as the quantum kicked rotor (QKR) [1] and the quantum walk (QW) [2], have exposed unexpected behaviors that suggest new challenges both theoretical and experimental in the field of quantum information processing [3]. The behavior of the QKR has two characteristic modalities: dynamical localization (DL) and ballistic spreading of the variance in resonance. These different behaviors depend on whether the period of the kick is a rational or irrational multiple of 4π. For rational multiples the behavior of the system is resonant and the average energy grows ballistically and for irrational multiples the average energy of the system grows, for a short time, in a diffusive manner and afterwards DL appears. Quantum resonance is a constructive interference phenomena and DL is a destructive one. The DL and the ballistic behavior have already been observed experimentally [4,5]. On the other hand the concept of QW introduced in [6,7] is a counterpart of the classical random walk. Its most striking property is its ability to spread over the line linearly in time, this means that the standard deviation grows as σ(t) ∼ t, while in the classical walk it grows as σ(t) ∼ t 1/2 . We have developed [8,9] a parallelism between the behavior of the QKR and a generalized form of the QW showing that these models have similar dynamics. In [10] we have investigated the resonances of the QKR subjected to an excitation that follows an aperiodic Fibonacci prescription; there we proved that the primary resonances retain their ballistic behavior while the secondary resonances show a sub-ballistic wave function spreading (σ(t) ∼ t c with 0.5 < c < 1) like the QW with the same prescription for the coin [11]. Casati et al. [12] have studied the dynamics of the QKR kicked according to a Fibonacci sequence outside the resonant regime, they found sub-diffusive behavior for small kicking strengths and a threshold above which the usual diffusion is recovered. More recently Schomerus and Lutz [13] investigated the QKR subjected to a Lévy noise [14] and they show that this decoherence never fully destroys the DL of the QKR but leads to a sub-diffusion regime * Corresponding author. E-mail address: alejo@fing.edu.uy for a short time before DL appears.In this article we investigate the QKR in resonant regime and the usual QW when both are subjected to decoherence with a Lévy noise. In the case of the QKR the model has two strength parameters whose action alternate in a such way that the time interval between them follows a power law distribution. In the case of QW the model uses two evolution operators whose alternation follows the same power law distribution. We show that this noise in the...

We study the resonances of the quantum kicked rotor subjected to an excitation that follows an aperiodic Fibonacci prescription. In such a case the secondary resonances show a sub-ballistic behavior like the quantum walk with the same aperiodic prescription for the coin. The principal resonances maintain the well-known ballistic behavior.

The exact expression of the one-dimensional Boltzmann multiple-scattering coefficients, for the passage of particles through a slab of a given material, is obtained in terms of the single-scattering cross section of the material, including absorption. The remarkable feature of the result is that for multiple scattering in a metal, free from absorption, one recovers the well-known Landauer result for conduction electrons. In the case of particles, such as neutrons, moving through a weak absorbing media, Landuer's formula is modified due to the absorption cross section. For photons, in a strong absorbing media, one recovers the Lambert-Beer equation. In this latter case one may therefore speak of absorption-controlled diffusive processes.

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