Abstract. We show how to extend (and with what limitations) Avila's global theory of analytic SL(2,C) cocycles to families of cocycles with singularities. This allows us to develop a strategy to determine the Lyapunov exponent for extended Harper's model, for all values of parameters and all irrational frequencies. In particular, this includes the self-dual regime for which even heuristic results did not previously exist in physics literature. The extension of Avila's global theory is also shown to imply continuous behavior of the LE on the space of analytic M 2 (C)-cocycles. This includes rational approximation of the frequency, which so far has not been available.

Nonlinear optical signals from an assembly of N noninteracting particles consist of an incoherent and a coherent component, whose magnitudes scale ~ N and ~ N(N − 1), respectively. A unified microscopic description of both types of signals is developed using a quantum electrodynamical (QED) treatment of the optical fields. Closed nonequilibrium Green's function expressions are derived that incorporate both stimulated and spontaneous processes. General (n + 1)-wave mixing experiments are discussed as an example of spontaneously generated signals. When performed on a single particle, such signals cannot be expressed in terms of the nth order polarization, as predicted by the semiclassical theory. Stimulated processes are shown to be purely incoherent in nature. Within the QED framework, heterodyne-detected wave mixing signals are simply viewed as incoherent stimulated emission, whereas homodyne signals are generated by coherent spontaneous emission.

Optical signals obtained by the material response to classical laser fields are given by nonlinear response functions which can be expressed by sums over various quantum pathways of matter. We show that some pathways can be selected by using nonclassical fields, through the entanglement of photon and material pathways, which results in a different-power law dependence on the incoming field intensity. Spectrally overlapping stimulated Raman scattering (SRS) and twophoton-absorption (TPA) pathways in a pump probe experiment are separated by controlling the degree of entanglement of pairs of incoming photons. Pathway-selectivity opens up new avenues for mapping photon into material entanglement. New material information, otherwise erased by interferences among pathways, is revealed.Entanglement is one of the most fascinating manifestations of quantum mechanics. Entangled particles act as a single object even when they are spatially well separated, thus forming a "Schrödinger cat". Measurements with entangled particles have been used to test for the foundations of quantum mechanics by resolving the Einstein-Podolsky-Rosen (EPR) paradox and demonstrating Bell's inequalities [1,2]. Many applications to quantum information processing, secure communication (teleportation) [3], and quantum computing are based on the manipulation of entangled particles [4,5]. Photon entanglement is easier to create and maintain by e.g. parametric down conversion [6] or bi-exciton decay [7,8] than that of material particles [9]. Most optical applications of entanglement tune the photon frequencies away from material resonances. Nonlinear coupling between optical modes is then described by an effective optical field interaction Hamiltonian. The matter plays only a passive role by providing the interaction parameters through its susceptibilities but its degrees of freedom do not actively participate in the process. A clear signature of entanglement in nonlinear optics is that the near-resonant sum frequency generation signal scales linearly (rather than quadratically) with the incoming field intensity. This effect which indicates that the two photons effectively act as a single particle [10,11,12] has been predicted and verified experimentally by several groups [13,14,15,16,17,18,19]. Improved interferometric resolution by entanglement has been demonstrated [25].Here we show that nonlinear resonant interactions between entangled photons and matter can lead to a much more dramatic effect; quantum pathway selection. In resonant processes the matter actively participates and gets entangled with the photons, making it possible to control the pathway of matter by manipulating the photons. Pathway interference may be controlled by varying the degree of entanglement, thereby improving the resolution of nonlinear spectroscopic techniques. NIH-PA Author ManuscriptNIH-PA Author Manuscript NIH-PA Author ManuscriptNonlinear optical signals induced by classical optical fields are given by sums of terms which represent different possible quantum ...

Abstract. The extended Harper's model, proposed by D.J. Thouless in 1983, generalizes the famous almost Mathieu operator, allowing for a wider range of lattice geometries (parametrized by three coupling parameters) by permitting 2D electrons to hop to both nearest and next nearest neighboring (NNN) lattice sites, while still exhibiting its characteristic symmetry (Aubry-André duality). Previous understanding of the spectral theory of this model was restricted to two dual regions of the parameter space, one of which is characterized by the positivity of the Lyapunov exponent. In this paper, we complete the picture with a description of the spectral measures over the entire remaining (self-dual) region, for all irrational values of the frequency parameter (the magnetic flux in the model). Most notably, we prove that in the entire interior of this regime, the model exhibits a collapse from purely ac spectrum to purely sc spectrum when the NNN interaction becomes symmetric. In physics literature, extensive numerical analysis had indicated such "spectral collapse," however so far not even a heuristic argument for this phenomenon could be provided. On the other hand, in the remaining part of the self-dual region, the spectral measures are singular continuous irrespective of such symmetry. The analysis requires some rather delicate number theoretic estimates, which ultimately depend on the solution of a problem posed by Erdős and Szekeres in [28].

We survey the theory of quasi-periodic Schrödinger-type operators, focusing on the advances made since the early 2000s by adopting a dynamical systems point of view.