The Floquet Solution for Systems with Quadratic Form Hamiltonians

Abstract

Periodic multilayer structures of poly(p-phenylenevinylene)s have been fabricated by a self-assembly method on quartz or indium-tin-oxide-coated glass substrates. Alternating multilayers consisting of poly(1,4-(2-(5-carboxypentyloxy)-5-methoxyphenylene)vinylene) and poly(p-phenylenevinylene) were adsorbed onto positively charged substrates. Periodic multilayers with a microdisk geometry have also been fabricated on quartz or indium-tin-oxide-coated glass substrates. Their optical properties have been studied, and a yellow electroluminescence from a light-emitting device with a microdisk geometry has been observed.

The Perturbative Floquet Solution for Quasi-free Electrons

“…In order to ¢nd the perturbed quasi-energies and the reduced Floquet eigenvectors, we perform the extension in the Floquet space [8], and we obtain a problem formally similar to the static perturbation problem of a conservative system [9]. The Floquet Hamiltonian of the system, in the extended space F , is…”

“…In the absence of the static, spatial-periodic ¢eld one can derive the exact solutions of the eigenvalue equation of the Floquet Hamiltonian (i.e. the quasi-energies and the reduced Floquet eigenvectors) [8]. Since the plane xOy is a phase surface of the electro-magnetic ¢eld, the common dipolar approximation is here an exact result.…”

“…Since the plane xOy is a phase surface of the electro-magnetic ¢eld, the common dipolar approximation is here an exact result. In the dipolar-electric gauge the Hamiltonian is ĥ0 t 1 2m 0 p2 x p2 y e 0 i 0 x cos ot ; 40 and the principal solutions of the eigenvalue equation of the Floquet Hamiltonian are, [8] e 0 k x k y T 2 =2m 0 k 2 x k 2 y e 0 ; 41 j j 0 k x k y s t i e i T n 0 sin ot e Àix 0 k x cos otÀ1 Â j v x k x Àk 0 sin ot i j v y k y i j w s s i ; 42 where x 0 e 0 i 0 =m 0 o 2 , Tk 0 e 0 i 0 =o, n 0 e 2 0 i 2 0 = 8m 0 o 3 , e 0 e 2 0 i 2 0 =4m 0 o 2 , j w s s i is the eigenvector of the spin operator corresponding to the eigenvalue s AE1, and j v a k a i is the eigenvector of the momentum operator on the Oa-direction corresponding to the eigenvalue Tk a . By imposing cyclic boundary conditions we get quantized eigenvalues k a 2p=L a l a , where l a 0; AE1; AE2; .…”

“…Recently, we have obtained the exact solution of the Floquet problem (i.e. the quasi-energy spectrum and the reduced Floquet eigenvectors) for a particle in 2-dimensional space interacting with a monochromatic electro-magnetic ¢eld (and, in addition, with a possible magneto-static or elastic ¢eld) [8]. In the present paper we use those results in studying a particle interacting with a strong time-dependent electromagnetic ¢eld and with a static space-periodic ¢eld of weak intensity.…”

Many-Body Fermion Systems in the Floquet Formalism

“…Next, for the self-consistency of the paper, we list the most important properties of the Floquet Hamiltonian to be used later (see also [3,5,14]). The solution corresponding to p 0 is the principal solution, [E 0 Na E Na is the principal quasi-energy of the N-particle system, and jF Na;0 ii jF Na ii is the principal reduced Floquet eigenvector].…”