  We propose a general quantum theory of optical phase and instantaneous
frequency in the time domain for slowly varying optical signals. Guided by
classical estimation theory, we design homodyne phase-locked loops that enable
quantum-limited measurements of temporal phase and instantaneous frequency.
Standard and Heisenberg quantum limits to such measurements are then derived.
For optical sensing applications, we propose multipass and Fabry-P\'erot
position and velocity sensors that take advantage of the signal-to-noise-ratio
enhancement effect of wideband angle modulation without requiring nonclassical
light. We also generalize our theory to three spatial dimensions for
nonrelativistic bosons and define an Hermitian fluid velocity operator, which
provides a theoretical underpinning to the current-algebra approach of quantum
hydrodynamics.
