Spin-$\frac{1}{2}$ particle in an absorbing environment

Abstract

The quantum dynamics of a localized spin-$\frac{1}{2} $ Particle interacting with an absorbing environment is investigated. The quantum Langevin-Schr\"odinger equation for spin-$\frac{1}{2} $ is obtained. The susceptibility function of the environment is calculated in terms of the coupling function of the spin and the environment. it is shown that the susceptibility function satisfies the Kramers-Kronig relations. Spontaneous emission and the shift frequency of the spin is obtained in terms of the imaginary part of the susceptibility function in frequency domain. Some transition probabilities between the spin states are calculated when the absorbing environment is in the thermal state.