Thesis (Ph. D.)--University of Rochester. Dept. of Physics and Astronomy, 2016.
We present a stochastic path integral formalism to study statistical behaviour of a
quantum system under weak continuous measurement. The path integral is constructed
from joint probability density functions of measurement outcomes and
quantum state trajectories. The optimal dynamics, such as the most likely path,
is obtained by extremizing the action of the stochastic path integral. We also
explore advantages of having the full joint probability distribution of quantum
trajectories by applying exact functional methods as well as developing a perturbative
approach to investigate the statistical behaviour of continuous quantum
measurement. Examples are given for qubits measured in the ${\hat \sigma}_z$ basis and qubits
undergoing fluorescence, where their most likely evolutions, average trajectories,
variances, and multi-time correlation functions are investigated. Moreover, we
verify the theoretical prediction for the most likely paths with experimental data
from superconducting transmon qubits coupled to microwave cavities. We present
the experiment in two different cases: one is a single transmon qubit continuously
measured in time, and another is two qubits that are jointly measured and spatially
separated in two microwave cavities. We show that the qubits' trajectories
can be accurately tracked, and the qubits' state statistics and optimal dynamics
can be predicted using our stochastic path integral and action principle approach.
