Action formalism for geometric phases from self-closing quantum trajectories

TL;DR

This paper investigates how the geometric phase of quantum trajectories in a qubit system undergoes a topological transition influenced by measurement strength, using a stochastic path integral approach to analyze rare self-closing events.

Contribution

It introduces a formalism combining stochastic path integrals with measurement-induced geometric phases to analyze topological transitions in quantum trajectories.

Findings

Geometric phase exhibits a topological transition as measurement strength varies.

Inclusion of Gaussian corrections shifts the transition point, matching numerical simulations.

Most likely trajectories determine the topological transition behavior.

Abstract

When subject to measurements, quantum systems evolve along stochastic quantum trajectories that can be naturally equipped with a geometric phase observable via a post-selection in a final projective measurement. When post-selecting the trajectories to form a close loop, the geometric phase undergoes a topological transition driven by the measurement strength. Here, we study the geometric phase of a subset of self-closing trajectories induced by a continuous Gaussian measurement of a single qubit system. We utilize a stochastic path integral that enables the analysis of rare self-closing events using action methods and develop the formalism to incorporate the measurement-induced geometric phase therein. We show that the geometric phase of the most likely trajectories undergoes a topological transition for self-closing trajectories as a function of the measurement strength parameter. Moreover, the inclusion of Gaussian corrections in the vicinity of the most probable self-closing trajectory quantitatively changes the transition point in agreement with results from numerical simulations of the full set of quantum trajectories.