In this paper, the moderately long-time behaviour of the correlation functions for two charged coupled harmonic oscillators connected to a common heat bath are analysed in the presence of a magnetic field via the quantum Langevin dynamics. Interestingly, it is seen that long-time correlation functions at \(T\rightarrow 0\) exhibit a power-law decay with coefficients of the power laws being completely different for the two masses, affecting the overall dynamics of the coupled system. The effect of the bath-induced force on mass \(m_1\) mediated by the interaction of \(m_2\) with the common heat bath is studied and the results are highlighted in the presence of an external magnetic field. It is shown that the effect of cyclotron frequency increases the magnitudes of correlation functions at an instant of time by lowering the rate of temporal decay in the presence of a uniform magnetic field. The results in the absence of magnetic field are also presented, which are extremely important for investigating the movements of atoms in a protein molecule at low temperature.
