Nonlinearly driven Landau-Zener transition in a qubit 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 upsilon and switching rate gamma. The qubit energy levels are driven nonlinearly in time, proportional to sgn(t)vertical bar t vertical bar(nu), and we derive the transition probability in the limit of sufficiently fast noise, for arbitrary exponent nu. The level occupation after the transition depends strongly on nu, and there exists a critical nu(c) with qualitative difference between nu nu(c). When 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 phase coherence, which is relevant when the qubit has a nonzero minimum energy gap. The qualitative dependency on nu is the same for the phase coherence and level occupation. The state after the transition does, in general, depend on gamma. For fixed upsilon, increasing gamma decreases the final state coherence when nu < 1 and increases the final state coherence when nu>1. Only the conventional linear driving is independent of gamma.