Localising Relational Degrees of Freedom in Quantum Mechanics

In this thesis I present a wide-ranging study of localising relational degrees of freedom, contributing to the wider debate on relationism in quantum mechanics. A set of analytical and numerical methods are developed and applied to a diverse range of physical systems. Chapter 2 looks at the interference of two optical modes with no prior phase correlation. Cases of initial mixed states — specifically Poissonian states and thermal states — are investigated in addition to the well-known case of initial Fock states. For the pure state case, and assuming an ideal setup, a “relational Schrödinger cat” state emerges localised at two values of the relative phase. Circumstances under which this type of state is destroyed are explained. When the apparatus is subject to instabilities, the states which emerge are sharply localised at one value. Such states are predicted to be long lived. It is shown that the localisation of the relative phase can be as good, and as rapid, for initially mixed states as for the pure state case. Chapter 3 extends the programme of the previous chapter discussing a variety of topics — the case of asymmetric initial states with intensity very much greater in one mode, the transitive properties of the localising process, some applications to quantum state engineering (in particular for creating large photon number states), and finally, a relational perspective on superselection rules. Chapter 4 considers the spatial interference of independently prepared BoseEinstein condensates, an area which has attracted much attention since the work of Javanainen and Yoo. The localisation of the relative atomic phase plays a key role here, and it is shown that the phase localises much faster than is intimated in earlier studies looking at the emergence of a well-defined pattern of interference. A novel analytical method is used, and the predicted localisation is compared with the output of a full numerical simulation. The chapter ends with a review of a related body of literature concerned with non-destructive measurement of relative atomic phases between condensates. Chapter 5 explores localising relative positions between mirrors or particles scattering light, addressing recent work by Rau, Dunningham and Burnett. The analysis here retains the models of scattering introduced by those authors but makes different assumptions. Detailed results are presented for the case of free particles, initially in thermal states, scattering monochromatic light and thermal light. It is assumed that an observer registers whether or not an incident light packet has been scattered into a large angle, but lacks access to more detailed information. Under these conditions the localisation is found to be only partial, regardless of the number of observations, and at variance with the sharp localisation reported previously.