  We propose a quantum harmonic oscillator measurement engine fueled by
simultaneous quantum measurements of the non-commuting position and momentum
quadratures of the quantum oscillator. The engine extracts work by moving the
harmonic trap suddenly, conditioned on the measurement outcomes. We present two
protocols for work extraction, respectively based on single-shot and
time-continuous quantum measurements. In the single-shot limit, the oscillator
is measured in a coherent state basis; the measurement adds an average of one
quantum of energy to the oscillator, which is then extracted in the feedback
step. In the time-continuous limit, continuous weak quantum measurements of
both position and momentum of the quantum oscillator result in a coherent
state, whose coordinates diffuse in time. We relate the extractable work to the
noise added by quadrature measurements, and present exact results for the work
distribution at arbitrary finite time. Both protocols can achieve unit work
conversion efficiency in principle.
