= Solution
The individual rates describe <population relaxation>: $\gamma_{12}$ transfers population from level one to level two, and $\gamma_{21}$ transfers it back. With $p_1+p_2=1$ and the conventional qubit coordinate $r_z=p_1-p_2$,
$$
\dot p_1=-\gamma_{12}p_1+\gamma_{21}p_2,\qquad \dot r_z=-(\gamma_{12}+\gamma_{21})r_z+(\gamma_{21}-\gamma_{12}).
$$
Thus $T_1^{-1}=\gamma_{12}+\gamma_{21}$ is the longitudinal relaxation rate. The parameter $\Gamma=T_2^{-1}$ is the <transverse relaxation> rate: it damps the off-diagonal coherence, or the $x,y$ components of the <Bloch vector>. In the usual completely positive two-level model,
$$
\Gamma=\frac12(\gamma_{12}+\gamma_{21})+\gamma_\phi,\qquad\gamma_\phi\geq0,
$$
so $\Gamma$ includes both the coherence loss caused by <population relaxation> and additional pure <dephasing>.
There is a normalization switch in the displayed qubit <Affine Bloch equation>. Its constant term is appropriate to the conventional <Pauli matrices> and unit-radius coordinate $r_k=\operatorname{Tr}(\rho\sigma_k^{\mathrm{Pauli}})$. With the orthonormal generators used earlier, $s=r/\sqrt2$, the same physical rates would instead give a constant term $(\gamma_{21}-\gamma_{12})/\sqrt2$. The drift matrix is unchanged. We interpret the supplied qubit equation in its conventional $r$ coordinates.
With zero controls and $g=\gamma_{12}+\gamma_{21}>0$, $\Gamma>0$, the <steady state> is unique:
$$
\boxed{r_*=(0,0,(\gamma_{21}-\gamma_{12})/g),\qquad \rho_* =\operatorname{diag}(\gamma_{21}/g,\gamma_{12}/g).}
$$
The earlier normalized vector is $s_*=r_*/\sqrt2$. Indeed $r_x,r_y$ decay as $e^{-\Gamma t}$ and $r_z-r_{*,z}$ as $e^{-gt}$.
For vanishing rates, the full equilibrium conditions of the <Affine Bloch equation> are $\Gamma r_x=\Gamma r_y=0$ and $g r_z=\gamma_{21}-\gamma_{12}$, together with $\|r\|\leq1$. In particular, for nonnegative rates, $g=0$ means both population rates vanish. If then $\Gamma>0$, all diagonal <density operators> are stationary; if also $\Gamma=0$, every <density operator> is stationary. The formal case $g>0$, $\Gamma=0$ permits extra transverse equilibrium coordinates, but is excluded by the completely positive rate bound above.
Back to article page
