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 comparison of the physiological demands of rugby union match-play when determined by absolute (ABS) or individual (IND) velocity bands in Global Positioning System (GPS) analysis

Since turning professional in 1995 there have been considerable advances in the research on the demands of rugby union, largely using Global Positioning System (GPS) analysis over the last 10 years. A systematic review on the use of GPS, particularly the setting of absolute (ABS) and individual (IND) velocity bands in field based, intermittent, high-intensity (HI) team sports was undertaken. From 3669 records identified, 38 studies were included for qualitative analysis. Little agreement on the definition of movement intensities within team sports was found, only three papers, all on rugby union, had used IND bands, with only one comparing ABS and IND methods. Thus, the aim of this study was to determine if there is a difference in the demands within positions when comparing ABS and IND methods for GPS analysis and if these differences are significantly different between the forward and back positional groups. A total of 214 data files were recorded from 26 players in 17 matches of the 2015/2016 Scottish BT Premiership. ABS velocity zones 1-7 were set at 1) 0-6, 2) 6.1-11, 3) 11.1-15, 4) 15.1-18, 5) 18.1-21, 6) 21.1-15 and 7) 25.1-40km.h-1 while IND zones 1-7 were 1) <20, 2) 20-40, 3) 40-50, 4) 50-70, 5) 70-80, 6) 80-95 and 7) 95-100% of player’s individually determined maximum velocity (Vmax). A 40m sprint test measured Vmax using OptaPro S4 10 Hz (catapult, Australia) GPS units to derive IND bands. The same GPS units were worn during matches. GPS outputs analysed were % distance, % time, high intensity efforts (HIEs) over 18.1 km.h-1 / 70% max velocity and repeated high intensity efforts (RHIEs) which consists of three HIEs in 21secs. General linear model (GLM) analysis identified a significant difference in the measurement of % total distance covered, between the ABS and IND methods in all zones for forwards (p<0.05) and backs (p<0.05). This difference was also significant between forwards and backs in zones 1, shown as mean difference ± standard deviation (3.7±0.7%), 6 (1.2±0.4%) and 7 (1.0±0.0%) respectively (p<0.05). Percentage time estimations were significantly different between ABS and IND analysis within forwards in zones 1 (1.7±1.7%), 2 (-2.9±1.3%), 3 (1.9±0.8%), 4 (-1.4±0.8%) and 5 (0.2±0.4%), and within backs in zones 1 (-10±1.5%), 2 (-1.2±1.1%), 3 (1.8±0.9%) and 5 (0.6±0.5%) (p<0.05). The difference between groups was significant in zones 1, 2, 4 and 5 (p<0.05). The number of HIEs was significantly different between forwards and backs in zones 6 (6±2) and 7 (3±2). RHIEs were significantly different between ABS and IND for forwards (1±2, p<0.05) although not between groups. Until more research on the differences in ABS and IND methods is carried out, then neither can be deemed a criterion method. In conclusion, there are significant differences between the ABS and IND methods of GPS analysis of the physical demands of rugby union, which must be considered when used to inform training load and recovery to improve performance and reduce injuries.