  We derive a non-Markovian master equation for a charged particle in a
magnetic field coupled to a bath and study decoherence by analysing the
temporal decay of the off-diagonal elements of the reduced density matrix in
the position basis. The coherent oscillations characterised by the cyclotron
frequency get suppressed as a result of decoherence due to coupling with the
environment. We consider an Ohmic bath with three distinct models for the
high-frequency cutoff for the spectral density of the bath and compare the
three cases. As expected, the three cutoff models converge in the limit of the
uppermost frequency of the bath tending to infinity. We notice a dramatic
slowing down of loss of coherence in the low-temperature limit dominated by
zero point quantum fluctuations compared to the high-temperature classical
limit dominated by thermal fluctuations. We also go beyond the Ohmic model and
study super-Ohmic and sub-Ohmic baths with the spectral densities deviating
from a linear dependence on the frequency. Our results are testable in a state
of the art cold atom laboratory.
