We use a discrete approximation of quantum mechanics called the quantum random walk to study diffusion in one-dimensional crystalline nanostructures. There, intense fluctuations in the density are a consequence of quantum interference. As the size of the crystal increase, relative quantum fluctuations decrease and, if a coarse-grained averaging is taken, interference may be neglected. When this happens, we describe a far-from-equilibrium, classical, dynamic mesoscopic diffusion regime. In this regime, diffusion equations are given by a persistent-random-walk process. Density becomes a second-order Markov process, and in the continuum limit, the current satisfies the non-Fickian Maxwell-Cattaneo relationship. The generalized diffusion equation in the telegraphist's equation.
With the help of quantum-scattering-theory methods and the approximation of the stationary phase, we propose a one-dimensional quantum-random-walk (QRW) model, which describes for both tunneling and scattering above the potential, the coherent diffusion of independent particles described by wave packets in a periodic one-dimensional lattice. The QRW model describes for each lattice cell the time evolution of modulating amplitudes of two opposite-moving wave packets as they are scattered by periodic potential barriers. Since the QRW model is a coherent process, interference contributions in the probabilities bring about strong departures from classical results. For many identical free particles we obtain the theoretical and graphical Bose and Fermi two-body QRW probability distribution. The result is generalized to N identical free particles and we obtain the N-body Bose and Fermi QRW probability distribution.
Classical random walks have found applications ranging from Brownian motion, diffusion of particles, designing computer algorithms, and even in the study of biological systems.Random walks can be used to explain the process of diffusion in the driftdiffusion model charge carrier motion in semiconductors.In the classical treatment of transport phenomena, the charge carrier transport can be characterized by a well defined diffusion coefficient D. Recently, there have been numerous proposals to make electronic and logic devices based on the spin property of electron like the Datta-Das Spin Field Effect Transistor.In such devices, the transport is non-classical (ballistic).This document examines the possibility of defining a spin diffusion coefficient for nonclassical transport, using the study of quantum random walks as a starting point.In addition, this document also analyzes the effect of uniform noise on the behavior of quantum random walks and driving the quantum walk into classical regime.i Boolchand for his constant motivation and support throughout my stay at the University of Cincinnati and his valuable suggestions.I would like to thank Dr. Altan Ferendeci, who was on my Master's defense committee, for his suggestions to improve the quality of this work.I extend my thanks to Dr. Rohana Wijewardhana in the Physics department of University of Cincinnati for allowing me to attend his lectures.I thank my lab mates Maitreya Datta and Nikhil Bhandari for their encouragement and the stimulating environment in the lab.Special thanks are in order to my multitude of friends, those in Cincinnati for making my stay here enjoyable and those elsewhere for their support, encouragement and belief in
We study discrete-time quantum walks with Fourier, Grover, and Hadamard coin operators on disordered square lattices, alongside classical random walks. Edges in the lattice are absent with a probability,
p. In this case, both classical and quantum walks exhibit an essentially identical degradation of transport. We then study a general version of this model, in which the existing edges are retained. However, the absent edges are no longer absent, but now become partially transmitting, with a transmission coefficient
f. We find that the response is now quite sensitive to the walk protocol and coin symmetry. The classical walk exhibits asymptotic stability of the anomalous diffusion exponent,
β(t). In comparison, quantum walks exhibit a monotonically decaying
β(t) beyond the initial transient regime, leading to the absence of quantum diffusion in two dimensions. We map the transport regimes in the
(f,p) plane and identify the contour for
β=1, which separates the subdiffusive and superdiffusive regimes. We also examine the effective diffusion coefficient, the inverse participation ratio, and the first two moments of edge transmission. We restrict ourselves to analyzing bulk propagation prior to finite-size boundary reflections. The results indicate that quantum transport depends significantly on coin-specific interference and cannot be fully described by transmission statistics alone.
In this work, we theoretically investigate a one-dimensional bosonic and fermionic discrete in time quantum Walk, affected by a temporal disorder. Specifically, we study how the wave-function symmetry and the phase disorder distribution affect the final coincidence probability and transport properties of a 100-step quantum Walk. The degree of randomness is suitably adjusted by the disorder distribution width and different diffusion regimes are achieved, ranging from the classical diffusion to the ballistic, associated with quantum states propagation.
We generalize the quantum random walk protocol for a particle in a one-dimensional chain, by using several types of biased quantum coins, arranged in aperiodic sequences, in a manner that leads to a rich variety of possible wave-function evolutions. Quasiperiodic sequences, following the Fibonacci prescription, are of particular interest, leading to a sub-ballistic wave-function spreading. In contrast, random sequences lead to diffusive spreading, similar to the classical random walk behavior. We also describe how to experimentally implement these aperiodic sequences.