Pedagogical model for transverse relaxation and motional narrowing of magnetic resonacne spectroscopy

A standard relaxation theory known as the BPP theory is of common use in magnetic resonance spectroscopy. A typical phenomenon that can be briefly elucidated by the theory is a broadening of peaks in Fourier spectra due to a transverse relaxation of a spin magnetization. There are, however, drawbacks: this theory is a classical phenomeno- logical theory and often claimed to be insufficient to describe small relaxation time. In addition, it cannot describe a motional narrowing without introducing a phenomenological approach. We introduce a simple model in which a system spin ? ? is coupled with an environmental system localized around the system. The environmental system probabilistically dissipates into a thermal bath and a thermal environmental system migrates into the place instead. We show analytically that this model can describe a transverse relaxation and a motional narrowing simultaneously. We also show numerical simulations of decoherence in a spin system coupled with on-resonance bosonic modes. Our model will hopefully contribute to easier understanding of motional narrowing for a pedagogical purpose.