Nonlinear Landau-Zener-St\"uckelberg-Majorana problem

TL;DR

This paper extends the Landau-Zener-St"uckelberg-Majorana model to include nonlinear parameter variations, deriving analytic transition probabilities for perturbative nonlinearities and analyzing the validity of the DDP formula through numerical comparisons.

Contribution

It introduces analytic expressions for transition probabilities in nonlinear LZSM problems and assesses the validity of the DDP formula for both perturbative and essential nonlinearities.

Findings

Perturbative nonlinearities yield accurate analytic transition probabilities.

The DDP formula remains valid under small nonlinear corrections.

Essential nonlinearities can significantly alter the transition dynamics.

Abstract

In the standard Landau-Zener-St\"uckelberg-Majorana (LZSM) problem, the bias sweep rate and gap are both time independent and fully characterize the LZSM problem. We consider the nonlinear LZSM problem, in which at least one of the two characteristic parameters varies as the system traverses the avoided crossing region. This situation results in what could be thought of as a more accurate description of any realistic situation as compared to the idealized linear LZSM problem. We consider both the case of perturbative nonlinearities, where the nonlinearity adds small corrections to the linear problem, and the case of essential nonlinearities, where the sweep and/or minimum-gap functions are qualitatively different from those of the linear LZSM problem. In the case of perturbative nonlinearities, we derive analytic expressions for the LZSM transition probability based on the Dykhne-Davis-Pechukas (DDP) formula, taking into account the leading corrections to the standard LZSM formula. We compare the derived approximate expressions with numerical simulation results and comment on the validity of the approximations. In particular, if the nonlinear term is small in comparison to the linear term throughout the finite duration of the avoided crossing traversal, the perturbative approximation is valid. Our results also provide information about the validity of the DDP formula. In addition to reviewing cases of essential nonlinearity treated previously in the literature, we analyze the case of an essentially nonlinear sweep function that describes an almost square pulse.