Nonlinearly driven Landau-Zener transition in a qubit 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 $\ensuremath{\upsilon}$ and switching rate $\ensuremath{\gamma}$. The qubit energy levels are driven nonlinearly in time, $\ensuremath{\propto}\mathrm{sgn}(t){\ensuremath{\mid}t\ensuremath{\mid}}^{\ensuremath{\nu}}$, and we derive the transition probability in the limit of sufficiently fast noise, for arbitrary exponent $\ensuremath{\nu}$. The level occupation after the transition depends strongly on $\ensuremath{\nu}$, and there exists a critical ${\ensuremath{\nu}}_{c}$ with qualitative difference between $\ensuremath{\nu}<{\ensuremath{\nu}}_{c}$ and $\ensuremath{\nu}>{\ensuremath{\nu}}_{c}$. When $\ensuremath{\nu}<{\ensuremath{\nu}}_{c}$, the final state is always fully incoherent with equal population of both quantum levels, even for arbitrarily weak noise. For $\ensuremath{\nu}>{\ensuremath{\nu}}_{c}$, 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 ${\ensuremath{\nu}}_{c}=1∕2$, while for slow noise ${\ensuremath{\nu}}_{c}<1∕2$ and it depends on $\ensuremath{\gamma}$. We also discuss phase coherence, which is relevant when the qubit has a nonzero minimum energy gap. The qualitative dependency on $\ensuremath{\nu}$ is the same for the phase coherence and level occupation. The state after the transition does, in general, depend on $\ensuremath{\gamma}$. For fixed $\ensuremath{\upsilon}$, increasing $\ensuremath{\gamma}$ decreases the final state coherence when $\ensuremath{\nu}<1$ and increases the final state coherence when $\ensuremath{\nu}>1$. Only the conventional linear driving is independent of $\ensuremath{\gamma}$.