Heller’s expression for the absorption cross-section in the weak field limit is extended to cases where the total Hamiltonian contains a strong time-dependent component, supplemented by a weak field. A very similar expression to the original case then results when the (t,t) formalism is used; one only needs to construct a correlation function for the system without the weak field, and use it to extract the absorption probability for any value of the weak-field frequency (or pulse shape). In addition, a numerical approach for extracting Floquet states without full-matrix diagonalization is demonstrated, by filtering (or filter-diagonalization) a single wave function (or the correlation function) propagated under the (t,t) Hamiltonian.
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As an additional check on the approach, we have also done the following check (due to a suggestion by S. H. Patil). We define a modified the initial wave function as simply , where the purpose of the is to modify the wave function so that it would vanish at and at . Throughout the propagation of X it should still be true then that , since in principle every ray corresponds to a different solution of the time-dependent Schrödinger equation, and the solution at the two rays [associated with and ] is explicitly made to have a vanishing amplitude at the start of the rays (at ). Indeed, a numerical propagation of X under was found to fulfill this property to better than
26.
The energy spectrum in the Floquet problem is infinitely large (and periodic) so that the choice of the energy levels by their order in the region is arbitrary (and not necessarily related to the zero-field order). Thus, energy levels “0” and “9” are really neighboring in energies, which leads to the corresponding peak in Figure 1. The peaks in the graph are similar when 14 basis functions are used.
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