A nonlinear Landau-Zener model was proposed recently to describe, among a
number of applications, the nonadiabatic transition of a Bose-Einstein
condensate between Bloch bands. Numerical analysis revealed a striking
phenomenon that tunneling occurs even in the adiabatic limit as the nonlinear
parameter $C$ is above a critical value equal to the gap $V$ of avoided
crossing of the two levels. In this paper, we present analytical results that
give quantitative account of the breakdown of adiabaticity by mapping this
quantum nonlinear model into a classical Josephson Hamiltonian. In the critical
region, we find a power-law scaling of the nonadiabatic transition probability
as a function of $C/V-1$ and $\alpha $, the crossing rate of the energy levels.
In the subcritical regime, the transition probability still follows an
exponential law but with the exponent changed by the nonlinear effect. For
$C/V>>1$, we find a near unit probability for the transition between the
adiabatic levels for all values of the crossing rate.Comment: 9 figure
We investigate the interaction of a two-dimensional model atom with an intense, high-frequency circularly polarized laser pulse. As the laser intensity is increased, the ionization rate initially increases, then decreases dramatically, with the electron wave function developing an asymmetric ring form which rotates with the electric field. We provide evidence that this wave form is due to localization of the electron onto nonlinear classical structures.
By writing the complete set of 3 + 1 (ADM) equations for linearized waves, we are able to demonstrate the properties of the initial data and of the evolution of a wave problem set by Alcubierre and Schutz. We show that the gauge modes and constraint error modes arise in a straightforward way in the analysis, and are of a form which will be controlled in any well specified convergent computational discretization of the differential equations.
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