  We present a formulation of measurement-based feedback control of a single
quantum particle in one spatial dimension. An arbitrary linear combination of
the position and momentum of the particle is continuously monitored, and
feedback proportional to the measured signal is used to control the system. We
derive a feedback master equation and discuss a general approach to computing
the steady-state solutions for arbitrary potentials. For a quantum harmonic
oscillator or a free particle, we show that it is possible to cool and confine
the system using feedback that simultaneously damps the measured observable and
its conjugate momentum, as well as compensates for noise introduced by the
measurement. In addition, we demonstrate that appropriate feedback adds a
quadratic term in the measured observable to the Hamiltonian of the system. For
the particular case of the harmonic potential, we describe the exact dynamics
of the system that can be cooled to the ground state. Moreover, we provide an
argument for the possibility to cool systems with arbitrary potentials provided
that the measurement is strong enough to localise the particle on an interval
smaller than the characteristic length scale of the potential.

Quantum Physics