Effects of nonlinear sweep in the Landau-Zener-Stueckelberg effect

We study the Landau-Zener-Stueckelberg (LZS) effect for a two-level system with a time-dependent nonlinear bias field (the sweep function) $W\left(t\right).$ Our main concern is to investigate the influence of the nonlinearity of $W\left(t\right)\mathrm{}$ on the probability P to remain in the initial state. The dimensionless quantity $\varepsilon =\pi {\Delta }^2/\left(2\mathrm{}ħv\right)$ depends on the coupling $\Delta$ of both levels and on the sweep rate $v.\mathrm{}$ For fast sweep rates, i.e., ɛ≪1, and monotonic, analytic sweep functions linearizable in the vicinity of the resonance we find the transition probability $1-P\cong \varepsilon \mathrm{}(1+a),$ where $a>\mathrm{}0\mathrm{}$ is the correction to the LSZ result due to the nonlinearity of the sweep. Further increase of the sweep rate with nonlinearity fixed brings the system into the nonlinear-sweep regime characterized by $1-P\cong {\varepsilon }^{\gamma }$ with γ≠1, depending on the type of sweep function. In the case of slow sweep rates, i.e., ɛ≫1, an interesting interference phenomenon occurs. For analytic $W\left(t\right)\mathrm{}$ the probability ${P=P}_0{e}^{-\eta }$ is determined by the singularities of $\sqrt{{\Delta }^2{+W}^2\mathrm{}\left(t\right)\mathrm{}}$ in the upper complex plane of t. If $W\left(t\right)\mathrm{}$ is close to linear, there is only one singularity, which leads to the LZS result ${P=e}^{-\varepsilon }$ with important corrections to the exponent due to nonlinearity. However, for, e.g., $W\left(t\right)\mathrm{}\propto t^3$ there is a pair of singularities in the upper complex plane. Interference of their contributions leads to oscillations of the prefactor $P_0$ that depends on the sweep rate through ɛ and turns to zero at some ɛ. Measurements of the oscillation period and of the exponential factor would allow one to determine Δ, independently.