  We present a protocol to produce a class of non-thermal Fock state mixtures
in trapped ions. This class of states features a clear metrological advantage
with respect to the ground state, thus overcoming the standard quantum limit
without the need for full sideband cooling and Fock-state preparation on a
narrow electronic transition. The protocol consists in the cyclic repetition of
red-sideband, measurement and preparation laser pulses. By means of the Kraus
map representation of the protocol, it is possible to relate the length of the
red sideband pulses to the specific class of states that can be generated. With
the help of numerical simulations, we analyze the parametric regime where these
states can be reliably reproduced.
