  We use the Quantum Langevin equation as a starting point to study the
response function, the position-velocity correlation function and the velocity
autocorrelation function of a charged Quantum Brownian particle in the presence
of a magnetic field and linearly coupled to a heat bath via position
coordinate. We study two bath models -- the Ohmic bath model and the Drude bath
model -- and make a detailed comparison in various time-temperature regimes.
For both bath models there is a competition between the cyclotron frequency and
the viscous damping rate giving rise to a transition from an oscillatory to a
monotonic behaviour as the damping rate is increased. In the zero point
fluctuation dominated low temperature regime, non-trivial noise correlations
lead to some interesting features in this transition. We study the role of the
memory time scale which comes into play in the Drude model and study the effect
of this additional time scale. We discuss the experimental implications of our
analysis in the context of experiments in cold ions.
