Universal spectral properties of spatially periodic quantum systems with chaotic classical dynamics (original) (raw)
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Weakly chaotic attractors of an intermittent map defined on the complex unit circle arise in an analogy with a recurrence relation of the scattering matrix associated with wave transport through locally periodic structures of consecutive sizes. It is demonstrated that the fixed-point solution (infinite iteration time or scattering structure size) of the relation corresponds to an average of the scattering matrix over a set, or ''ensemble'', of systems of all sizes. This ergodic property implies the analyticity of the scattering matrix S and the existence of its ''ensemble'' average ⟨S⟩, called the optical S-matrix. We find that the invariant density of the map that governs the sample-to-sample fluctuations of coherent transport is given by the Poisson kernel of potential theory, and consequently the distribution of S is uniquely determined by ⟨S⟩ which depends only on the transport properties of a single scattering cell. The theoretical distribution, closely related to the Cauchy distribution, shows perfect agreement with numerical results for a chain of delta potentials. A consequence of our findings is the a priori knowledge of ⟨S⟩ without the customary resort to experimental data.