  The effect of atom losses on a homogeneous one-dimensional Bose gas lying
within the quasi-condensate regime is investigated using a Monte Carlo
wavefunction approach. The evolution of the system is calculated, conditioned
by the loss sequence, namely the times of individual losses and the position of
the removed atoms. We describe the gas within the linearized Bogoliubov
approach. For each mode, we find that, for a given quantum trajectory, the
state of the system converges towards a coherent state, i.e. the ground state,
displaced in phase space. Provided losses are recorded with a temporal and
spatially resolved detector, we show that quantum feedback can be implemented
and cooling to the ground state of one or several modes can be realized.
