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, $\propto \sign(t)|t|^\nu$, and we derive the transition probability in the limit of sufficiently fast noise, for arbitrary exponent $\nu$. The longitudinal coherence after transition depends strongly on $\nu$, and there exists a critical $\nu_c$ with qualitative difference between $\nu< \nu_c$ and $\nu > \nu_c$. When $\nu<\nu_c$ the end state is always fully incoherent with equal population of both quantum levels, even for arbitrarily weak noise. For $\nu>\nu_c$ 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 $\nu_c=1/2$, while for slow noise $\nu_c<1/2$ 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 $\nu$ 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 $\nu<1$ and increase the final state coherence when $\nu>1$. Only the conventional linear driving is independent of $\gamma$.