Quantum trajectory theory, developed largely in the quantum optics community to describe open quantum systems subjected to continuous monitoring, has applications in many areas of quantum physics. In this paper I present a simple model, using two-level quantum systems (q-bits), to illustrate the essential physics of quantum trajectories and how different monitoring schemes correspond to different ``unravelings'' of a mixed state master equation. I also comment briefly on the relationship of the theory to the Consistent Histories formalism and to spontaneous collapse models.
Quantum open systems are described in the Markovian limit by master equations in Lindblad form. I argue that common ``quantum trajectory'' techniques representing continuous measurement schemes, which solve the master equation by unravelling its evolution into stochastic trajectories in Hilbert space, correspond closely to particular sets of decoherent (or consistent) histories. This is illustrated by a simple model of photon counting. An equivalence is shown for these models between standard quantum jumps and the orthogonal jumps of Di\'osi, which have already been shown to correspond to decoherent histories. This correspondence is compared to simple treatments of trajectories based on repeated or continuous measurements.
We introduce jump-time unraveling as a distinct description of open quantum systems. As our starting point, we consider quantum jump trajectories, which emerge, physically, from continuous quantum measurements, or, formally, from the unraveling of Markovian quantum master equations. If the stochastically evolving quantum trajectories are ensemble-averaged at specific times, the resulting quantum states are solutions to the associated quantum master equation. We demonstrate that quantum trajectories can also be ensemble-averaged at specific jump counts. The resulting jump-time-averaged quantum states are then solutions to a discrete, deterministic evolution equation, with time replaced by the jump count. This jump-time evolution represents a trace-preserving quantum dynamical map if and only if the associated open system does not exhibit dark states. In the presence of dark states, on the other hand, jump-time-averaged states may decay into the dark states and the jump-time evolution may eventually terminate. Jump-time-averaged quantum states and the associated jump-time evolution are operationally accessible in continuous measurement schemes, when quantum jumps are detected and used to trigger the readout measurements. We illustrate the jump-time evolution with the examples of a two-level system undergoing relaxation or dephasing, a damped harmonic oscillator, and a free particle exposed to collisional decoherence.
Quantum trajectories provide a fundamental description of the measurement of individual quantum systems. As such, they have wide impact, and application, in the emerging field of quantum technology. Importantly, they also give a mechanism for developing our understanding of the nature of quantum mechanics. \n \nIn Part I of this thesis, we develop and apply quantum trajectory methods, with a focus upon experimentally relevant optomechanical systems. By solving the stochastic master equation for sufficiently simple bosonic systems, and subsequently finding the positive operator-valued measure (POVM), we are able to conduct a detailed study of the use of parametric amplification for quantum state tomography of nonclassical optomechanical states of motion. Homodyne tomography is a cornerstone experimental tool, and an analysis of its convergence is carried out in the presence of realistic imperfections. We complete Part I by conducting a detailed preparatory analysis of superfluid optomechanical systems possessing vorticity. \n \nPart II of this thesis investigates the correspondence between open classical and open quantum systems. We prove a result shows that open quantum systems are, in general, harder to track than open classical systems. We couch this result in terms of physically realisable ensembles (PREs), which can describe the dynamics of a monitored,
D-dimensional, quantum system obeying a master equation that has reached equilibrium. Associated with the PRE is a measurement scheme that leads to quantum trajectories in which the system evolution consists purely of jumps between the states that are members of the PRE. The occupation of the
K, generally non-orthogonal, states in the ensemble can be used to track the system. The number of states in the ensemble,
K, represents the amount of memory that is required to do so. In comparison, a classical
D-dimensional system requires occupation of the
D states to be tracked. After first developing analysis tools that make feasible the discovery of PREs in
D>2, we prove our main result that there are quantum systems that have a minimal sized PRE with
K>D.
We review the continuous monitoring of a qubit through its spontaneous emission, at an introductory level. Contemporary experiments have been able to collect the fluorescence of an artificial atom in a cavity and transmission line, and then make measurements of that emission to obtain diffusive quantum trajectories in the qubit's state. We give a straightforward theoretical overview of such scenarios, using a framework based on Kraus operators derived from a Bayesian update concept; we apply this flexible framework across common types of measurements including photodetection, homodyne, and heterodyne monitoring and illustrate its equivalence to the stochastic master equation formalism throughout. Special emphasis is given to homodyne (phase-sensitive) monitoring of fluorescence. The examples we develop are used to illustrate basic methods in quantum trajectories, but also to introduce some more advanced topics of contemporary interest, including the arrow of time in quantum measurement, and trajectories following optimal measurement records derived from a variational principle. The derivations we perform lead directly from the development of a simple model to an understanding of recent experimental results.