Quantum theory and applications

点击翻译按钮获取中文摘要

Quantum mechanics governs essentially all phenomena at microscopic scales and below. I intend to examine several problems in quantum mechanics and its more sophisticated relative, quantum field theory. My work can be thought of as being in two fields: quantum field theory and applications, and quantum information (the former weighted more heavily than the latter). I am working on three projects falling roughly in the category of quantum field theory. The first is a field theoretic description of a phenomenon known as the Josephson effect, which gives rise to current flow between two superconductors separated by an insulator. This description is interesting because it can easily be generalized to a more complicated setting (one with what is known as a non-Abelian order parameter), as colleagues and I have demonstrated. I intend to deepen our understanding of this description, and to find applications of the non-Abelian effect to real physical systems. The second is a study of the possible phases in a toy field theory known as the Abelian Higgs model, with the addition of a so-called Chern-Simons term. (A "toy" model is one that is studied not because it describes a realistic physical system, but rather because it is simpler than realistic models, so it is easier to analyze mathematically, the hope being that such an analysis will shed light on more realistic models. In our case, the toy model is one that lives in only two space dimensions.) Studying the phase structure of a model is interesting because different phases have very different properties; one example is the confinement transition in the model of interactions among quarks (the constituents of protons, neutrons, and other less familiar particles). The third project is to continue studying the effect of topological defects on false vacuum decay. A false vacuum is one that is stable classically but unstable via tunnelling quantum mechanically. While this might seem far removed from reality, it is actually much like what happens in any phase transition, and particularly in the phase transitions that were known to occur in the early universe. I have worked on this problem in the recent past, and there is much still to be done. Two projects study quantum mechanical issues, with an eye towards quantum information. In one, I am studying the validity of an approximation in quantum mechanics known as the adiabatic approximation, which applies to time-dependent, yet slowly-evolving, systems. Controlling the reaction of the system to time-dependent external effects is the name of the game. This control is necessary almost across the board in quantum mechanics, but it is particularly at the forefront in quantum information, where manipulation of qubits (quantum-mechanical units of information) is arguably the most important task to master. The second project is to study spin lattices as a tool for investigation the foundations of quantum mechanics, and specifically Bell's Theorem. While the first of these is an area I have worked on for several years, the second is something I have begun to think about only recently. The rough idea is to consider a finite lattice of spins, the usual protagonists Alice and Bob having access to spins at the ends of a spin chain or ladder. In this model, the intermediate spins play the role of hidden variables, and the objective is to study correlations between measurements made by Alice and Bob in this setting.