Ions stored in Penning traps may have useful applications in the field of
quantum information processing. There are, however, difficulties associated
with the laser cooling of one of the radial motions of ions in these traps,
namely the magnetron motion. The application of a small radio-frequency
quadrupolar electric potential resonant with the sum of the two radial motional
frequencies has been shown to couple these motions and to lead to more
efficient laser cooling. We present an analytical model that enables us to
determine laser cooling rates in the presence of such an 'axializing' field. It
is found that this field leads to an averaging of the laser cooling rates for
the two motions and hence improves the overall laser cooling efficiency. The
model also predicts shifts in the motional frequencies due to the axializing
field that are in qualitative agreement with those measured in recent
experiments. It is possible to determine laser cooling rates experimentally by
studying the phase response of the cooled ions to a near resonant excitation
field. Using the model developed in this paper, we study the expected phase
response when an axializing field is present.Comment: 22 pages, 7 figure