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.

A distributed, compact routing protocol for the Internet

The Internet has grown in size at rapid rates since BGP records began, and continues to do so. This has raised concerns about the scalability of the current BGP routing system, as the routing state at each router in a shortest-path routing protocol will grow at a supra-linearly rate as the network grows. The concerns are that the memory capacity of routers will not be able to keep up with demands, and that the growth of the Internet will become ever more cramped as more and more of the world seeks the benefits of being connected. Compact routing schemes, where the routing state grows only sub-linearly relative to the growth of the network, could solve this problem and ensure that router memory would not be a bottleneck to Internet growth. These schemes trade away shortest-path routing for scalable memory state, by allowing some paths to have a certain amount of bounded “stretch”. The most promising such scheme is Cowen Routing, which can provide scalable, compact routing state for Internet routing, while still providing shortest-path routing to nearly all other nodes, with only slightly stretched paths to a very small subset of the network. Currently, there is no fully distributed form of Cowen Routing that would be practical for the Internet. This dissertation describes a fully distributed and compact protocol for Cowen routing, using the k-core graph decomposition. Previous compact routing work showed the k-core graph decomposition is useful for Cowen Routing on the Internet, but no distributed form existed. This dissertation gives a distributed k-core algorithm optimised to be efficient on dynamic graphs, along with with proofs of its correctness. The performance and efficiency of this distributed k-core algorithm is evaluated on large, Internet AS graphs, with excellent results. This dissertation then goes on to describe a fully distributed and compact Cowen Routing protocol. This protocol being comprised of a landmark selection process for Cowen Routing using the k-core algorithm, with mechanisms to ensure compact state at all times, including at bootstrap; a local cluster routing process, with mechanisms for policy application and control of cluster sizes, ensuring again that state can remain compact at all times; and a landmark routing process is described with a prioritisation mechanism for announcements that ensures compact state at all times.