We show that particle transport, as characterized by the equilibrium mean square displacement, in a uniform, quantum multibaker map, is generically ballistic in the long time limit, for any fixed value of Planck's constant. However, for fixed times, the semiclassical limit leads to diffusion. Random matrix theory provides explicit analytical predictions for the mean square displacement of a particle in the system. These results exhibit a crossover from diffusive to ballistic motion, with crossover time on the order of the inverse of Planck's constant. We expect that, for a large class of 1D quantum random walks similar to the quantum multibaker, a sufficient condition for diffusion in the semiclassical limit is classically chaotic dynamics in each cell. The systems described generalize known quantum random walks and may have applications for quantum computation.
We show that particle transport, as characterized by the equilibrium mean square displacement, in a uniform, quantum multibaker map, is generically ballistic in the long time limit, for any fixed value of Planck's constant. However, for fixed times, the semiclassical limit leads to diffusion. Random matrix theory provides explicit analytical predictions for the mean square displacement of a particle in the system. These results exhibit a crossover from diffusive to ballistic motion, with crossover time on the order of the inverse of Planck's constant. We expect that, for a large class of 1D quantum random walks similar to the quantum multibaker, a sufficient condition for diffusion in the semiclassical limit is classically chaotic dynamics in each cell. The systems described generalize known quantum random walks and may have applications for quantum computation.
The common perception is that strong coupling to the environment will always render the evolution of the system density matrix quasiclassical (in fact, diffusive) in the long time limit. We present here a counterexample, in which a particle makes quantum transitions between the sites of a d-dimensional hypercubic lattice while strongly coupled to a bath of two-level systems that ``record'' the transitions. The long-time evolution of an initial wave packet is found to be most unusual: the mean square displacement of the particle density matrix shows long-range ballistic behavior, with $⟨{n}^{2}⟩\ensuremath{\sim}{t}^{2}$, but simultaneously a kind of weakly localized behavior near the origin. This result may have important implications for the design of quantum computing algorithms, since it describes a class of quantum walks.
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 introduce a family of models for quantum mechanical, one-dimensional random walks, called quantum multibaker maps (QMB). These are Weyl quantizations of the classical multibaker models previously considered by Gaspard, Tasaki, and others. Depending on the properties of the phase's parametrizing the quantization, we consider only two classes of the QMB maps: uniform and random. Uniform QMB maps are characterized by phases that are the same in every unit cell of the multibaker chain. Random QMB maps have phases that vary randomly from unit cell to unit cell. The eigenstates in the former case are extended while in the latter they are localized. In the uniform case and for large variant Planck's over 2pi, analytic solutions can be obtained for the time-dependent quantum states for periodic chains and for open chains with absorbing boundary conditions. Steady state solutions and the properties of the relaxation to a steady state for a uniform QMB chain in contact with "particle" reservoirs can also be described analytically. The analytical results are consistent with, and confirmed by, results obtained from numerical methods. We report here results for the deep quantum regime (large variant Planck's over 2pi ) of the uniform QMB, as well as some results for the random QMB. We leave the moderate and small variant Planck's over 2pi results as well as further consideration of the other versions of the QMB for further publications.
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.
We study the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain. In the unitary (i.e., nonrandom) case, we allow any unitary operator which commutes with translations and couples only sites at a finite distance from each other. For example, a single step of the walk could be composed of any finite succession of different shift and coin operations in the usual sense, with any lattice dimension and coin dimension. We find ballistic scaling and establish a direct method for computing the asymptotic distribution of position divided by time, namely as the distribution of the discrete time analog of the group velocity. In the random case, we let a Markov chain (control process) pick in each step one of finitely many unitary walks, in the sense described above. In ballistic order, we find a nonrandom drift which depends only on the mean of the control process and not on the initial state. In diffusive scaling, the limiting distribution is asymptotically Gaussian, with a covariance matrix (diffusion matrix) depending on momentum. The diffusion matrix depends not only on the mean but also on the transition rates of the control process. In the nonrandom limit, i.e., when the coins chosen are all very close or the transition rates of the control process are small, leading to long intervals of ballistic evolution, the diffusion matrix diverges. Our method is based on spatial Fourier transforms, and the first and second order perturbation theory of the eigenvalue 1 of the transition operator for each value of the momentum.
Quantum walks are quantum counterparts of random walks. While probability distributions of random walks diffusively spread out as the walkers are updating, quantum walks have ballistic behavior. Some of the ballistic behaviors have been revealed in long-time limit theorems and their probability distributions are all far away from the Gaussian distributions, which are known as limit distributions of random walks. In this chapter, we are going to be seeing limit distributions for a standard quantum walk on the line and two kinds of time-dependent quantum walk on the line.