Floquet theory and continued fractions for harmonically driven systems
Abstract
We derive an exact solution using continued fractions for a quantum particle
scattering from an oscillating delta-function potential. We study its transmission
properties such as: Transmission zeros, transmission poles and threshold
anomalies. Using the same technique and a translation matrix method, we
study the problem of an infinite chain of oscillating deltas. We calculate its
band structure and eigenstates and show explicitly the contribution to these
eigenstates from the quasi-bound state of a single oscillating delta. We study
the dynamics of the quasi-energy bands of the system as a function of the
strength of the oscillation and show band quasi-periodicity and band collapse.
We also define the Floquet-Green’s function for a time-periodic Hamiltonian
and by a generalization of the method used for the two previous potentials we
are able to derive an expression for the Floquet-Green’s function of any harmonically
driven Hamiltonian. As an example of the application of this method
we study a tight-binding Hamiltonian with harmonic time dependence.
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