Quantum Trajectories and Their Extremal-Probability Paths: New Phenomena and Applications

Abstract

Quantum theory posits that measurements are invasive: It describes an interactive process, mediating all knowledge of quantum properties, in which an observer probes a quantum system and draws inferences about its behavior based on the outcomes they receive. This dissertation emphasizes the theory, and several applications, of time-continuous monitoring of quantum systems. The particular results described below follow two threads: One concerns the behavior of extremal-probability trajectories, derived from a path integral description and action-extremization principle applied to diffusive quantum trajectories. The other concerns the use of continuous measurement for generating and preserving entanglement between two remote qubits.

Two new phenomena arising in the optimal paths, derived as the first--order variational solutions of a stochastic action, are presented herein. Multiple solutions which extremize the probability of the measurement record may connect the same initial and final quantum state over the same time interval; these groups of multipath solutions arise as dynamical instabilities in the optimal path dynamics, and in close mathematical analogy with the formation of caustics in optics. Furthermore, the proliferation of these quantum caustics is closely connected to chaotic behavior among extremal-probability paths. This optimal path chaos that we characterize fundamentally relies on measurement, and can occur in a system with no immediate classical analog, such as a qubit; this is in contrast with typical explorations of quantum chaos.

The other thread developed herein concerns methods to generate and protect entanglement between two remote qubits by joint continuous measurement of their spontaneous emission. We characterize the entanglement dynamics arising in quantum jump trajectories due to continuous photon counting, as well as in diffusive trajectories arising from homodyne monitoring. These processes generate equivalent entanglement yield on average, despite qualitatively different dynamics in their respective individual realizations. Finally, some strategies for feedback control based on these measurements are developed, with the goal of increasing the entanglement yield and lifetime. Specifically, we illustrate how implement a measurement un-doing and control procedure, that traps the two-qubit dynamics in limit cycles around Bell states for each type of measurements mentioned.