A Classical Perspective on Non-Diffractive Disorder
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CitationKlales, Anna. 2016. A Classical Perspective on Non-Diffractive Disorder. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences.
AbstractThe unifying themes connecting the chapters in this dissertation are the profound and often surprising effects of disorder in classical and quantum systems and the tremendous insight gained from a classical perspective, even in quantum systems. In particular, we investigate disorder in the form of weak, spatially correlated random potentials, i.e. far from the Anderson Localization regime.
We present a new scar-like phenomenon in quantum wells. With the introduction of local impurities to the oscillator, the eigenstates localize onto classical periodic orbits of the unperturbed system. Compared to traditional scars in chaotic billiards, these scars are both more common and stronger. Though the unperturbed system has circular symmetry, the random perturbation selects a small number of orientations which are shared by many scarred states -- dozens or even hundreds -- over a range of energies. We show, via degenerate perturbation theory, that the cause of the new scars is the combination of an underlying classical resonance of the unperturbed system and a perturbation induced coupling that is strongly local in action space.
Next we examine the same type of local perturbation applied to an open system: branched flow. Caustics in the manifold of trajectories have been implicated in the formation of strong branches. We show that caustic formation is intimately tied to compression of manifolds of trajectories in phase space, which has important implications for the position space density.
We introduce the "Kick and Drift" model, a generalization of the standard map. The model is a good approximation to the full two dimensional dynamics of a wave propagating over a weak random potential, but it provides a simpler framework for studying branched flow.
Next we develop a classical model for electrons executing cyclotron motion in a graphene flake and implement it numerically. We derive classical equations of motion for electrons moving through the graphene flake with a position dependent effective mass due to fluctuations in the background carrier density. I apply these methods to an experiment performed by the Westervelt group. They imaged the flow of electrons in a graphene flake by measuring the transresistance as they rastered a charged scanning probe microscope tip over the surface. My simulations show that the regions with the greatest change in transresistance do always coincide with the regions with the highest current density. Furthermore I show that the experimental results can qualitatively reproduced by treating the system classically.
Finally, we extend Heller's thawed Gaussian approximation from second order in the classical action to third order, in order to capture curvature in phase space. Such phase space dynamics are ubiquitous in systems with weak random potentials, such as those discussed above. We derive a closed form solution, but find that more work needs to be done to make it numerically tractable and competitive with other methods. A semiclassical method capturing phase space curvature could provide insight into the behavior of scars away from the hbar goes to zero limit.
Citable link to this pagehttp://nrs.harvard.edu/urn-3:HUL.InstRepos:26718765
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