This dissertation covers several projects making use of weak quantum measurement. Measuring a property of a quantum system is an invasive process that disturbs the system, so our knowledge of the system is inherently limited. When the strength of the measurement decreases, there is less disturbance to the system, but less information is gained; this tradeoff is a theme throughout the work presented here. The first three projects presented here cover applications of quantum trajectories, where by making many sequential weak measurements, it is possible to obtain a stochastic record of the system evolving over time. The final project is a proposal for an integrated device using weak value amplification to sensitively measure optical frequency.First, we propose using quantum trajectories to reconstruct an unknown time-dependent Hamiltonian. Existing techniques either rely on projective measurements of the system before and after coherent time evolution and do not explicitly reconstruct the full time-dependent Hamiltonian or interrupt evolution for tomography. In contrast to previous work, our technique does not require interruptions, which would distort the recovered Hamiltonian. We introduce a novel algorithm which recovers the Hamiltonian and density matrix from an incomplete set of continuous measurements and demonstrate that it reliably extracts amplitudes of a variety of single-qubit and entangling two-qubit Hamiltonians.Next, we analyze the continuous monitoring of a many-level extension of a qubit, called a qudit, coupled to a cavity and read out using both phase-preserving and phase-sensitive amplification. The quantum trajectories of the system are described by a stochastic master equation, for which we derive the appropriate Lindblad operators. The measurement back-action causes spiraling in the state coordinates during collapse, which increases as the system levels become less distinguishable. We discuss two examples: a two-level system, and a $d$-dimensional system and meter with rotational symmetry in the quadrature space. We also provide a comparison of the effects of phase-preserving and phase-sensitive detection on the master equation and show that the average behavior is the same in both cases, but individual trajectories collapse at different rates depending on the measurement axis in the quadrature plane.Third, we demonstrate that continuously monitored quantum dynamics can be chaotic. The optimal paths between past and future boundary conditions can diverge exponentially in time when there is time-dependent evolution and continuous weak monitoring. We investigate the onset of chaos in pure-state qubit systems with optimal paths generated by a periodic Hamiltonian. Specifically, chaotic quantum dynamics are demonstrated in a scheme where two non-commuting observables of a qubit are continuously monitored, and one measurement strength is periodically modulated.Finally, we present an integrated design to sensitively measure changes in optical frequency using weak value amplification with a multi-mode interferometer. The technique involves introducing a weak perturbation to the system and then post-selecting the data in such a way that the signal is amplified without amplifying the technical noise, as has previously been demonstrated in a free-space setup. We demonstrate the advantages of a Bragg grating with two band gaps for obtaining simultaneous, stable high transmission and high dispersion. The device is more robust and easily scalable than the free-space implementation, and provides amplified sensitivity compared to other methods of measuring changes in optical frequency on a chip, such as an integrated Mach-Zehnder interferometer.
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