Nonlinearly driven Landau-Zener transition with telegraph noise

Abstract

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, $γ$. The qubit energy levels are driven nonlinearly in time, $\propto \sign(t)|t|^ν$, and we derive the transition probability in the limit of sufficiently fast noise, for arbitrary exponent $ν$. The longitudinal coherence after transition depends strongly on $ν$, and there exists a critical $ν_c$ with qualitative difference between $ν< ν_c$ and $ν> ν_c$. When $ν<ν_c$ the end state is always fully incoherent with equal population of both quantum levels, even for arbitrarily weak noise. For $ν>ν_c$ 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 $ν_c=1/2$, while for slow noise $ν_c<1/2$ and it depends on $γ$. We also discuss transverse coherence, which is relevant when the qubit has a nonzero minimum energy gap. The qualitative dependency on $ν$ is the same for transverse as for longitudinal coherence. The state after transition does in general depend on $γ$. For fixed $v$, increasing $γ$ decreases the final state coherence when $ν<1$ and increase the final state coherence when $ν>1$. Only the conventional linear driving is independent of $γ$.