Zhu-Nakamura theory and the superparabolic level-glancing models

Abstract

We study the applicability of the Zhu-Nakamura theory to a class of time-dependent quantum mechanical level-crossing models called superparabolic level-glancing models. The phenomenon of a level glancing, being on the borderline between a proper crossing of energy levels and an avoided crossing, is also an important special case between the two different approximative expressions in the Zhu-Nakamura theory. It is seen that the application of the theory to these models is not straightforward. We discuss some possible causes of these difficulties and also compare the approximative formulas of Zhu-Nakamura theory to those obtained by the generalization of the DDP theory.Comment: 10 pages, 3 figure

Analytic solutions of the quantum two-state problem in terms of the double, bi- and triconfluent Heun functions

“…Thirty-five five-parametric and fifteen four-parametric classes of models allowing solutions in terms of these functions have been derived, a useful feature of which is the extension of the previously known detuning modulation functionstwo-parametric at most -to functions involving more parameters. In the case of constant detuning this leads to two-peak symmetric or asymmetric pulses with controllable width [25], and, in the general case of variable detuning, it provides a variety of level-crossing models including symmetric and asymmetric models of non-linear sweeping through the resonance [32,33], level-glancing configurations [34][35][36], processes with two resonance-crossing time points [36] and multiple (periodically repeated) crossing models [37][38][39][40].…”

Two-level parabolic model with phase-jump coupling

“…The approximation P0 depends only on the values of the diabatic energy and coupling in one point of time. For example, if we consider more general superparabolic models [18,25] given by α ref (t) = t 2n − c where n = 1, 2, 4, . .…”

Geometry of adiabatic Hamiltonians for two-level quantum systems

“…This is due to fact that we usually have t i = −t f = −∞ while the adiabatic coupling, and therefore the corresponding curvature, differ appreciably from zero only near avoided crossings (which are usually chosen to happen near t = 0) and so the curves have well-defined lines as their asymptotes. Figure 1 depicts the situation for Landau-Zener [6,7,8,9] and parabolic [14,15,16] models. It also shows whether or not the model swaps the labels between the diabatic and adiabatic states, as discussed in connection with (11).…”

“…The latter, of course, gives the eigenenergies of the Hamiltonian and also the adiabatic coupling is directly read from these expressions via (16). The corresponding quantities are plotted in figure 2.…”