Nonlinearly driven Landau-Zener transition with telegraph noise

We study Landau-Zener like dynamics of a qubit influenced by transverse random telegraph noise. The telegraph noise is characterized by its coupling strength, $v$ and switching rate, $gamma$. The qubit energy levels are driven nonlinearly in time, $propto sign(t)|t|^ u$, and we derive the transition probability in the limit of sufficiently fast noise, for arbitrary exponent $ u$. The longitudinal coherence after transition depends strongly on $ u$, and there exists a critical $ u_c$ with qualitative difference between $ u< u_c$ and $ u > u_c$. When $ u< u_c$ the end state is always fully incoherent with equal population of both quantum levels, even for arbitrarily weak noise. For $ u> u_c$ the system keeps some coherence depending on the strength of the noise, and in the limit of weak noise no transition takes place. For fast noise $ u_c=1/2$, while for slow noise $ u_c<1/2$ and it depends on $gamma$. We also discuss transverse coherence, which is relevant when the qubit has a nonzero minimum energy gap. The qualitative dependency on $ u$ is the same for transverse as for longitudinal coherence. The state after transition does in general depend on $gamma$. For fixed $v$, increasing $gamma$ decreases the final state coherence when $ u<1$ and increase the final state coherence when $ u>1$. Only the conventional linear driving is independent of $gamma$.