Renormalization analysis of correlation properties in a quasiperiodically forced two-level system

Abstract

We give a rigorous renormalization analysis of the self-similarity of correlation functions in a quasiperiodically forced two-level system. More precisely, the system considered is a quantum two-level system in a time-dependent field consisting of periodic kicks with amplitude given by a discontinuous modulation function driven in a quasiperiodic manner at golden mean frequency. Mathematically, our analysis consists of a description of all piecewise-constant periodic orbits of an additive functional recurrence. We further establish a criterion for such orbits to be globally bounded functions. In a particular example, previously only treated numerically, we further calculate explicitly the asymptotic height of the main peaks in the correlation function.

Golden mean renormalization for a generalized Harper equation: The Ketoja–Satija orchid

“…Kuznetsov et al 23 have given a renormalization analysis of the onset of a strange nonchaotic attractor. 25,26 The existence of a strong coupling fixed point for the Harper equation in the case of flux = ͑ ͱ a 2 +4−a͒ / 2 satisfying 2 + a =1 (i.e., in continued fraction notation = ͓a , a , ...͔), has recently been established, 5 generalizing the results in Ref. 23 can be generalized and put on a rigorous foundation.…”

“…Note that the same renormalization equation, (2.18), appears in the analysis by Kuznetsov et al 23 of the onset of a strange nonchaotic attractor in quasiperiodically forced nonlinear systems, and in the study by Feudel et al 7 of correlations on a strange nonchaotic attractor. 26. 25 In this case piecewise-constant orbits of the recurrence are desired, as they are in an analysis of correlations in a barrier billiard.…”

Self-similar correlations in a barrier billiard

“…An additive version of (1.1), which may be thought of as a functional (quantised) version of the classical Fibonacci recursion, is the governing recursion in this case. In [12] we have given a complete description of the periodic orbits of this recursion thereby explaining and generalising the results in [6].…”

Fixed points of composition sum operators

“…We will show that, apart from a small set of CSOs which admit polynomials as fixed points, there are no analytic non-zero fixed points of an affine CSO in G. All other non-zero fixed points therefore have singularities of some sort. For some CSOs with real coefficients the singularities can be discontinuities (see [10] for an example), but in the context of CSOs defined on spaces of analytic functions (with singularities) it is logarithmic and unbounded isolated singularities which are of greatest interest, and we will analyse these below.…”

“…Whether or not T has a polynomial fixed point depends on the precise values of the a i and α i . Indeed, for a polynomial p(x) = p 0 +p 1 x+· · ·+p m x m , p m = 0, it is clear that T p is a polynomial of degree at most m. Inspecting the coefficient of x m in T p(x) = p(x), we have a 1 s m 1 + a 2 s m 2 + · · · + a s m = 1 (10) which is clearly a necessary condition for a polynomial fixed point of degree m. Conversely, suppose that (10) holds. Then if p(x) is of degree m, T p is of degree at most m − 1, and so T is degenerate and has non-trivial kernel.…”