Dynamics of the Two-State System with Ohmic Dissipation

Abstract

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