Decoherence and thermalization of Unruh-DeWitt detector in arbitrary dimensions

We study the decoherence and thermalization of an Unruh-DeWitt detector linearly coupled to the free massless scalar field in flat spacetime of arbitrary dimensions ($d\geq 2$). The initial state of the detector is chosen to be a pure state consisting of a linear superposition of ground and excited states, and we calculate the time evolution of reduced density matrix of the detector. Using perturbation method, we analytically derive the transition rate of the detector (the rate of change of the diagonal elements in the density matrix) and the decoherence rate (the rate of change of the off-diagonal elements in the density matrix). We find that the results are not the same in odd and even dimensional spacetimes, but the unitarity of the qubit is preserved in both cases. The real part of the decoherence rate is related to the transition rate, while the imaginary part may contain different forms of divergence terms in different dimensions due to the temporal order product operator and the singularities of the Wightman function for quantum field theory. We derive the recurrence formula to obtain the divergence terms in each dimension and analyze the renormalization problem.