Dynamics of the Two-State System with Ohmic Dissipation

The authors study a symmetric two-state system with "bare" tunneling frequency ${\Delta }_0$, dimensionless Ohmic dissipation coefficient $\alpha$, and heat-bath cutoff ${\omega }_c$. Defining ${\Delta }_r\equiv {\Delta }_0{\left(\frac{{\Delta }_0}{{\omega }_c}\right)}^{\frac{\alpha }{(1-\alpha )}}$ for $\alpha <1\mathrm{}$ and ${\Delta }_r\equiv 0$ for $\alpha >1\mathrm{}$, they find to lowest order in $\frac{{\Delta }_r}{{\omega }_c}$, $\frac{\mathrm{kT}}{\hslash {\omega }_c}$ (a) for all $\alpha \mathrm{kT}\gg \hslash {\Delta }_r$, incoherent relaxation at a rate $\left(\frac{{{\Delta }_0}^2}{{\omega }_c}\right)\left(\frac{\sqrt{\pi }}{2}\right)\left[\frac{\Gamma \left(\alpha \right)}{\Gamma (\alpha +\frac{1}{2})}\right]\times {\left(\frac{\pi \mathrm{kT}}{\hslash {\omega }_c}\right)}^{2\alpha -1\mathrm{}}$; (b) for $T=0\mathrm{}$, $\frac{1}{2}<\alpha <1\mathrm{}$, incoherent relaxation at a rate $\sim {\Delta }_r$; and (c) for $T=0\mathrm{}$, $0<\mathrm{}\alpha <\frac{1}{2}$, damped oscillations with frequency $\sim {\Delta }_r$ and $Q$ factor $\frac{1}{2}cot\left[\frac{\left(\frac{\pi }{2}\right)\alpha }{(1-\alpha )}\right]$ plus a power-law background.