Which-path problem for one and two particles with two degrees of freedom and a relation between transverse spatial structure and group velocity of light

Quantum mechanics, optics and indeed any wave theory exhibits the phenomenon of interference. In this thesis we present two problems investigating interference due to indistinguishable alternatives and a mostly unrelated investigation into the free space propagation speed of light pulses in particular spatial modes. In chapter 1 we introduce the basic properties of the electromagnetic field needed for the subsequent chapters. In chapter 2 we review the properties of interference using the beam splitter and the Mach-Zehnder interferometer. In particular we review what happens when one of the paths of the interferometer is marked in some way so that the particle having traversed it contains information as to which path it went down (to be followed up in chapter 3) and we review Hong-Ou-Mandel interference at a beam splitter (to be followed up in chapter 5). In chapter 3 we present the first of the interference problems. This consists of a nested Mach-Zehnder interferometer in which each of the free space propagation segments are weakly marked by mirrors vibrating at different frequencies [1]. The original experiment drew the conclusions that the photons followed disconnected paths. We partition the description of the light in the interferometer according to the number of paths it contains which-way information about and reinterpret the results reported in [1] in terms of the interference of paths spatially connected from source to detector. In chapter 4 we briefly review optical angular momentum, entanglement and spontaneous parametric down conversion. These concepts feed into chapter 5 in which we present the second of the interference problems namely Hong-Ou-Mandel interference with particles possessing two degrees of freedom. We analyse the problem in terms of exchange symmetry for both boson and fermion pairs and show that the particle statistics at a beam splitter can be controlled for suitably chosen states. We propose an experimental test of these ideas using orbital angular momentum entangled photons. In chapter 6 we look at the effect that the transverse spatial structure of the mode that a pulse of light is excited in has on its group velocity. We show that the resulting group velocity is slower than the speed of light in vacuum for plane waves and that this reduction in the group velocity is related to the spread in the wave vectors required to create the transverse spatial structure. We present experimental results of the measurement of this slowing down using Hong-Ou-Mandel interference.

Essays on cultural and institutional dynamics in economic development using spatial analysis

This thesis seeks to research patterns of economic growth and development from a number of perspectives often resonated in the growth literature. By addressing themes about history, geography, institutions and culture the thesis is able to bring to bear a wide range of inter-related literatures and methodologies within a single content. Additionally, by targeting different administrative levels in its research design and approach, this thesis is also able to provide a comprehensive treatment of the economic growth dilemma from both cross-national and sub-national perspectives. The three chapters herein discuss economic development from two broad dimensions. The first of these chapters takes on the economic growth inquiry by attempting to incorporate cultural geography within a cross-country formal spatial econometric growth framework. By introducing the global cultural dynamics of languages and ethnic groups as spatial network mechanisms, this chapter is able to distinguish economic growth effects accruing from own-country productive efforts from those accruing from interconnections within a global productive network chain. From this, discussions and deductions about the implications for both developed and developing countries are made as regards potentials for gains and losses from such types and levels of productive integration. The second and third chapters take a different spin to the economic development inquiry. They both focus on economic activity in Africa, tackling the relevant issues from a geo-intersected dimension involving historic regional tribal homelands and modern national and subnational administrative territories. The second chapter specifically focuses on attempting to adopt historical channels to investigate the connection between national institutional quality and economic development in demarcated tribal homelands at the fringes of national African borders. The third chapter on the other hand focuses on looking closer at the effects of demarcations on economic activity. It particularly probes how different kinds of demarcation warranted by two different but very relevant classes of politico-economic players have affected economic activity quite distinguishably within the resulting subnational regions in Africa.