Spin relaxation and decoherence of two-level systems

We revisit the concepts of spin relaxation and spin decoherence of two-level (spin-$1∕2$) systems. From two toy models, we clarify two issues related to the spin relaxation and decoherence: (1) For an ensemble of two-level particles each subjected to a different environmental field, there exists an ensemble relaxation time ${T}_{1}^{*}$, which is fundamentally different from ${T}_{1}$. When the off-diagonal coupling of each particle is in a single mode with the same frequency but a random coupling strength, we show that ${T}_{1}^{*}$ is finite while the spin-relaxation time of a single-spin ${T}_{1}$ and the usual ensemble decoherence time ${T}_{2}^{*}$ are infinite. (2) For a two-level particle under only a random diagonal coupling, its relaxation time ${T}_{1}$ shall be infinite, but its decoherence time ${T}_{2}$ is finite.