Abstract
We study the dynamics of a two-level crossing model with a parabolic separation of the diabatic energies. The solutions are expressed in terms of the tri-confluent Heun equations—the generalization of the confluent hypergeometric equations. We obtain analytical approximations for the state populations in terms of Airy and Bessel functions. Applicable expressions are derived for a large part of the parameter space. We also provide simple formulas which connect local solution in different time regimes. The validity of the analytical approximations is shown by comparing them to numerical simulations.
Highlights
Level crossing models are crucial for the understanding of non-adiabatic transitions in physics, chemistry and biology [1]
The tunneling probability exp(−ζ) in the LandauZener model only depends on a single dimensionless parameter ζ ≡ 2π f 2/( |(F1 − F2)V|), where f is the coupling matrix element in the diabatic basis, F1 and F2 are the slopes of the intersecting diabatic potential curves, and V is the velocity of the perturbation variable, e.g., the relative collision velocity [8]
We studied the transition dynamics of the parabolic model — a two-state system subject to a quadratically detuning over an infinite time interval
Summary
Level crossing models are crucial for the understanding of non-adiabatic transitions in physics, chemistry and biology [1]. At around the same time, Stuckelberg derived a sophisticated tunneling formula based on analytical continuation of the semi-classical WKB solutions across the Stokes lines [5], and Majorana derived the transition probability formula independently using integral representation of the survival amplitude in connection with the dynamics of a spin1/2 in a time-varying magnetic field [6]. For cases in which the crossing points merge together as a result of external electric or magnetic fields, the Landau-Zener linearization fails, and the linear-dependence of diabatic energies should be replaced by a parabolic one [18]. The dynamics of the parabolic level-crossing model, probably unknown to most physicists, may be written in terms of the tri-confluent Heun function [41], which is derived from the general Heun function [42] by the coalescence of three finite regular singularities with infinity.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
