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Self-similar solutions of the coherent diffusion equation are derived and measured. The set of real similarity solutions is generalized by the introduction of a nonuniform phase surface, based on the elegant Gaussian modes of optical diffraction. In an experiment of light storage in a gas of diffusing atoms, a complex initial condition is imprinted, and its diffusion dynamics is monitored. The self-similarity of both the amplitude and the phase pattern is demonstrated, and an algebraic decay associated with the mode order is measured. Notably, as opposed to a regular diffusion spreading, a self-similar contraction of a special subset of the solutions is predicted and observed.
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A visualization scheme for quantum many-body wavefunctions is described, which we have termed qubism. Its main property is its recursivity: increasing the number of qubits reflects in an increase in the image resolution. Thus, the plots are typically fractal. As examples, we provide images for the ground states of commonly used Hamiltonians in condensed matter and cold atom physics, such as Heisenberg or ITF. Many features of the wavefunction, such as magnetization, correlations and criticality, can be visualized as properties of the images. In particular, factorizability can be easily spotted, and a way to estimate the entanglement entropy from the image is provided.
Coherent diffusion pertains to the motion of atomic dipoles experiencing frequent collisions in vapor while maintaining their coherence. Recent theoretical and experimental studies on the effect of coherent diffusion on key Raman processes, namely Raman spectroscopy, slow polariton propagation, and stored light, are reviewed in this Colloquium.
The method of self-similar factor approximants is completed by defining the approximants of odd orders, constructed from the power series with the largest term of an odd power. It is shown that the method provides good approximations for transcendental functions. In some cases, just a few terms in a power series make it possible to reconstruct a transcendental function exactly. Numerical convergence of the factor approximants is checked for several examples. A special attention is paid to the possibility of extrapolating the behavior of functions, with arguments tending to infinity, from the related asymptotic series at small arguments. Applications of the method are thoroughly illustrated by the examples of several functions, nonlinear differential equations, and anharmonic models.
Self-similarity in the network traffic has been studied from several aspects: both at the user side and at the network side there are many sources of the long range dependence. Recently some dynamical origins are also identified: the TCP adaptive congestion avoidance algorithm itself can produce chaotic and long range dependent throughput behavior, if the loss rate is very high. In this paper we show that there is a close connection between the static and dynamic origins of self-similarity: parallel TCPs can generate the self-similarity themselves, they can introduce heavily fluctuations into the background traffic and produce high effective loss rate causing a long range dependent TCP flow, however, the dropped packet ratio is low.
We study multifractal properties in the spectrum of effective time-independent Hamiltonians obtained using a perturbative method for a class of delta-kicked systems. The evolution operator in the time-dependent problem is factorized into an initial kick, an evolution dictated by a time-independent Hamiltonian, and a final kick. We have used the double kicked $SU(2)$ system and the kicked Harper model to study butterfly spectrum in the corresponding effective Hamiltonians. We have obtained a generic class of $SU(2)$ Hamiltonians showing self-similar spectrum. The statistics of the generalized fractal dimension is studied for a quantitative characterization of the spectra.
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