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arXiv:cond-mat/9912387 · Physical Review Letters

Anomalous Spreading of Power-Law Quantum Wave Packets

Fabrizio Lillo, Rosario N. Mantegna

Published Feb 7, 2000• 22 citations
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Abstract

We introduce power-law tail quantum wave packets. We show that they can be seen as eigenfunctions of a Hamiltonian with a physical potential. We prove that the free evolution of these packets presents an asymptotic decay of the maximum of the wave packets which is anomalous for an interval of the characterizing power-law exponent. We also prove that the number of finite moments of the wave packets is a conserved quantity during the evolution of the wave packet in the free space.

DOI
10.1103/physrevlett.84.1061

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Comment on "Anomalous Spreading of Power-Law Quantum Wave Packets"

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A family of wave packets with power law tails are employed to analyze the long time dependence of the corresponding probability density. The densities, associated to packets for free particles in the one-dimensional space, with sufficiently long tails decay asymptotically in time following a $t^{-β}$ law (with $β<1$), slower than the usual $t^{-1}$ one, including the particular case $(ln t)^2/t$ not correctly discussed in F. Lillo and R. N. Mantegna (Phys. Rev. Lett. 84, 1061 (2000)).

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We introduce power-law tail quantum wave packets. We show that they can be seen as eigenfunctions of a Hamiltonian with a physical potential. We prove that the free evolution of these packets presents an asymptotic decay of the maximum of the wave packets which is anomalous for an interval of the characterizing power-law exponent. We also prove that the number of finite moments of the wave packets is a conserved quantity during the evolution of the wave packet in the free space.
We study some properties of a class of quantum wave packets with power-law spatial tails analitically. Almost all packets of this class have the property that all the moments of the position operator are infinite. We prove that the free evolution of these packets presents an asymptotic decay of the wave packet maximum which is anomalous with respect to the behavior observed in the free evolution of customary quantum wave packets. We also study the temporal evolution of the tails and we show that the dominant term of the asymptotic expansion is conserved.
A family of wave packets with power law tails are employed to analyze the long time dependence of the corresponding probability density. The densities, associated to packets for free particles in the one-dimensional space, with sufficiently long tails decay asymptotically in time following a t−βt^{-β}t−β law (with β<1β<1β<1), slower than the usual t−1t^{-1}t−1 one, including the particular case (lnt)2/t(ln t)^2/t(lnt)2/t not correctly discussed in F. Lillo and R. N. Mantegna (Phys. Rev. Lett. 84, 1061 (2000)).
We investigate the wave-packet dynamics of the power-law bond disordered one-dimensional Anderson model with hopping amplitudes decreasing as ${H}_{nm}\ensuremath{\propto}{\ensuremath{\mid}n\ensuremath{-}m\ensuremath{\mid}}^{\ensuremath{-}\ensuremath{\alpha}}$. We consider the critical case $(\ensuremath{\alpha}=1)$. Using an exact diagonalization scheme on finite chains, we compute the participation moments of all stationary energy eigenstates as well as the spreading of an initially localized wave packet. The eigenstates multifractality is characterized by the set of fractal dimensions of the participation moments. The wave packet shows a diffusivelike spread developing a power-law tail and achieves a stationary nonuniform profile after reflecting at the chain boundaries. As a consequence, the time-dependent participation moments exhibit two distinct scaling regimes. We formulate a finite-size scaling hypothesis for the participation moments relating their scaling exponents to the ones governing the return probability and wave-function power-law decays.
We propose a setup for which a power-law decay is predicted to be observable for generic and realistic conditions. The system we study is very simple: a quantum wave packet initially prepared in a potential well with (i) tails asymptotically decaying like ∼ x −2 and (ii) an eigenvalues spectrum that shows a continuous part attached to the ground or equilibrium state. We analytically derive the asymptotic decay law from the spectral properties for generic, confined initial states. Our findings are supported by realistic numerical simulations for state-of-the-art expansion experiments with cold atoms.
A general relationship is established between wave-packet evolution and quantization in classically integrable systems. Because of wave-packet spreading, one cannot simply take the Fourier transform of the time evolution of a wave packet. Instead, one must propagate the wave packet using the actions as Hamiltonians. The energy eigenvalues which result are the Einstein-Brillouin-Keller values, and new forms for the eigenfunctions appear. These are free of caustic singularities, and represent averages of wave packets over the invariant torus.