We study the dynamics of a two-level crossing model with a cubic modification of the linear Landau-Zener tunneling, and express the non-adiabatic transition amplitudes in terms of the bi-confluent Heun functions which are generalizations of the confluent hypergeometric functions. We find a closed-form series expression of the transition probability at the long time limit, and derive tractable approximate formulas to the state populations by use of simple functions in a large part of the parameter space. The analytical approximations are validated by comparison with numerical results.
We study the dynamics of a nonlinear two-level crossing model with a cubic modification of the linear Landau-Zener diabatic energies. The solutions are expressed in terms of the bi-confluent Heun functions --- the generalization of the confluent hypergeometric functions. We express the finial transition probability as a convergent series of the parameters of the nonlinear laser detuning, and derive analytical approximations for the state populations in terms of parabolic cylinder and Whittaker functions. Tractable closed-form expressions are derived for a large part of the parameter space. We also provide simple method to determine the transition point which connects local solutions in different physical limits. The validity of the analytical approximations is shown by comparison with numerical results of simulations.
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.