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arXiv:1211.6356 · New Journal of Physics

Scale-free relaxation of a wave packet in a quantum well with power-law tails

Salvatore Miccichè, Andreas Buchleitner, Fabrizio Lillo, Rosario N. Mantegna, Tobias Paul, Sandro Wimberger

Published Mar 22, 2013• 6 citations
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Abstract

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.

DOI
10.1088/1367-2630/15/3/033033

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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.
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 analyze the wave-packet dynamics scenario produced by Random Matrix Theory models with unconventional band-profile structures. Our motivation is to understand the energy spreading of quantum systems with complex dynamics. Examples of such systems include complex nuclei, atoms and molecules, quantum dots, and Bose-Einstein Condensates in optical traps, which, under the influence of an external perturbation, experience an energy redistribution of the initially prepared state. Such a perturbation could be due to an external electric or magnetic field, a change in the confining geometry, or a residual interaction, among other things. Of special interest in our analysis is the investigation of the time relaxation properties of a prepared state into a sea of other states (the continuum). We find that, for a large family of power spectra characterized by a non-flat profile, the survival probability P(t)P(t)P(t) might exhibit either exponential-like or power-law decay.
Despite being ubiquitous, out-of-equilibrium quantum systems are much less understood than systems at equilibrium. Progress in the field has benefited from a symbiotic relationship between theoretical studies and new experiments on coherent dynamics. The present work strengthens this connection by providing a general picture of the relaxation process of isolated lattice many-body quantum systems that are routinely studied in experiments with cold atoms, ions traps, and nuclear magnetic resonance. We show numerically and analytically that the long-time decay of the probability for finding the system in its initial state necessarily shows a power-law behavior $\ensuremath{\propto}{t}^{\ensuremath{-}\ensuremath{\gamma}}$. This happens independently of the details of the system, such as integrability, level repulsion, and the presence or absence of disorder. Information about the spectrum, the structure of the initial state, and the number of particles that interact simultaneously is contained in the value of $\ensuremath{\gamma}$. From it, we can anticipate whether the initial state will or will not thermalize.
Despite being ubiquitous, out-of-equilibrium quantum systems are much less understood than systems at equilibrium. Progress in the field has benefited from a symbiotic relationship between theoretical studies and new experiments on coherent dynamics. The present work strengthens this connection by providing a general picture of the relaxation process of isolated lattice many-body quantum systems that are routinely studied in experiments with cold atoms, ions traps, and nuclear magnetic resonance. We show numerically and analytically that the long-time decay of the probability for finding the system in its initial state necessarily shows a power-law behavior ∝t−γ\propto t^{ - \gamma }∝t−γ. This happens independently of the details of the system, such as integrability, level repulsion, and the presence or absence of disorder. Information about the spectrum, the structure of the initial state, and the number of particles that interact simultaneously is contained in the value of γ\gammaγ. From it, we can anticipate whether the initial state will or will not thermalize.