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

TL;DR

This paper models the quantum dynamics of a spin-1/2 particle interacting with an absorbing environment, deriving equations and calculating properties like susceptibility, spontaneous emission, and transition probabilities.

Contribution

It introduces a quantum Langevin-Schrödinger equation for spin-1/2 particles in absorbing environments and relates susceptibility to the environment's coupling function.

Findings

Susceptibility function satisfies Kramers-Kronig relations

Spontaneous emission rate derived from susceptibility

Transition probabilities calculated in thermal 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.