Interplay of dissipation and memory in the quantum Langevin dynamics of a spin in a magnetic field

We derive a quantum Langevin equation for a quantum spin in the presence of a magnetic field and coupled to a bath. We have considered two models for the bath: an Ohmic bath model and a non-Markovian Drude bath model with finite memory. The effect of finite memory on the dynamics of the quantum spin is studied in detail. In particular, we have studied the effect of memory on the expectation values of the magnetic moment and the spin correlation functions. The time evolution study of the spin correlation functions shows that the correlation functions exhibit a damped oscillatory behavior with the randomization time being determined by the damping rate. We have analytically shown that the damping gets affected by the finite memory time when the bath retains memory of the interaction with the system. We also analyze the spin response function of the system for the Ohmic bath. Our results qualitatively agree with the findings in spin noise spectroscopy experiments and our predictions can be tested in future ultracold atom experiments.