Propriedades de transporte e emaranhamento em caminhadas quânticas em tempo discreto

Abstract

All the questions and situations presented in this thesis are complementary and intermingle to point the discussions towards its theme. We study the properties of transport, quantum entanglement, quantum coherence, superoscillations and the emergence of extreme events in discrete-time quantum walks (DTQWs). For this, it was necessary to insert some elements such as: aperiodicity, non-linearity, noises, disorder and artificial fields. We begin by studying how the localization properties, energy spectrum and quantum entanglement between degrees of freedom of a DTQWs are changed when adding an aperiodic spatial dependence in the quantum coin operator. The aperiodicity is described by coins spatially dependent on the positions in a 1D chain. Within the transport properties, we identified two regimes: delocalized and localized quantum walks mediated by an adequate adjustment of the parameter that controls the degree of aperiodicity of the distribution. Using energy spectra analysis, we show that at the initial stage inhomogeneity leads to a vanishing gap between two main bands, which justifies the predominantly delocalized character. For a sufficiently high degree of aperiodicity, we observe an energy spectrum, which resembles that described by Anderson’s one-dimensional model. Regarding the quantum entanglement of the system, we have shown many configurations where an increase in the ability to generate entanglement is observed. This behavior brings new information about the role of aperiodicity in this correlation between the space of position and the quantum coin for systems with static inhomogeneity, in contrast to what was previously reported, as almost always reducing entanglement when compared to the homogeneous case. Finally, we extend the analysis to show that systems with static heterogeneity are capable of exhibiting an asymptotic limit. Further on, inserting non-linearity, we study the existence and characterization of self-trapping phenomena in DTQWs. When considering a Kerr-type nonlinearity, we associate a probability density-dependent phase acquisition at each time step. Adjusting the nonlinear parameter and the quantum coins, we show the existence of different dynamic regimes, including those with traveling or self-trapping solitons-like structures. After mapping non-linear events in DTQWs, we propose a model to study the consequences of having noise and non-linear interaction associated with a qubit propagating in a circular chain. By employing quantum coherence measures, we report emerging unstable regimes in which quantum walks arise, such as self-focusing and breathing dynamics. Furthermore, we study the dynamics of a quantum walker submitted to independent and time-dependent phases simultaneously. Where such dynamics emulates a quantum particle charged in a lattice subjected to a superposition of static and harmonic electric fields. With appropriate adjustments, we investigated the possibility of inducing Bloch-type superoscillations, resulting from a tuning close to the harmonic phase frequency and that associated with Bloch-type oscillations. Furthermore, we show that under exact resonance conditions it is possible to establish a unidirectional motion. We show that the average drift velocity can be well described within an analog continuous-time model. Finally, we use the general framework of discrete-time quantum walks to study the physical origins of rogue waves through random phase modulation. We reveal its long-tail statistics, distribution profile, and dependence on the degree of randomness of the system. We were able to classify these extreme events as belonging to the Gumbel family of distributions.