Action Formalism for Measurement Induced Dynamics: Topological and Thermal Aspects

Shea, Dominic and Romito, Alessandro (2024) Action Formalism for Measurement Induced Dynamics: Topological and Thermal Aspects. PhD thesis, Lancaster University.

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Abstract

Feynman’s path integral is a formulation of quantum mechanics akin to analogous formulations developed for stochastic processes and statistical physics. One process that combines quantum dynamics with stochastic features and constitutes a prevailing area of research is quantum measurement. Beyond past attempts, recent advancements, exemplified by the Chantasari-Dressel-Jordan (CDJ) method, explore measurement-induced dynamics in continuously monitored quantum systems. This thesis utilises the CDJ path integral to explore new emerging features associated with measurement-induced dynamics. Firstly, we develop the CDJ path integral to analyse geometric phases. Focusing on self-closing trajectories from continuous measurements, we incorporate geometric phase information directly in the path integral for a single qubit and demonstrate that the geometric phase of the most likely trajectories exhibits a topological transition as a function of measurement strength. We further address the effect of Gaussian fluctuations. Secondly, we exploit the formalism to study measurement-induced entanglement dynamics, where we combine the stochastic effect of measurements and that of local unitary noise. We identify the optimal entanglement dynamics and develop diagrammatic methods that produce a closed-form approximation of the average entanglement dynamics. The optimal trajectories and diagrammatic expansion capture the oscillations of entanglement at short times. We find by numerical investigation that long-time steady-state entanglement reveals a non-monotonic relationship between concurrence and noise strength. Finally, we lay the basis for applying path integrals to fluctuation theorems by devising a suitable single qubit protocol to verify a recently proposed fluctuation theorem that governs the statistical behaviour of quantum systems far from equilibrium. The proposed protocol is suitable for existing quantum architectures. Our results provide a basis for extending the use of fluctuation theorems to many body systems, where geometric phases and entanglement play crucial roles in classifying quantum order.