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Quantum Langevin dynamics of a charged particle in a magnetic field : Response function, position-velocity and velocity autocorrelation functions

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.