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arXiv:0911.4694 · Physical Review Letters

Universal Response of Quantum Systems with Chaotic Dynamics

Diego A. Wisniacki, Natalia Ares, E. Vergini

Published Jun 23, 2010• 18 citations
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Abstract

The prediction of the response of a closed system to external perturbations is one of the central problems in quantum mechanics, and in this respect, the local density of states (LDOS) provides an in-depth description of such a response. The LDOS is the distribution of the overlaps squared connecting the set of eigenfunctions with the perturbed one. Here, we show that in the case of closed systems with classically chaotic dynamics, the LDOS is a Breit-Wigner distribution under very general perturbations of arbitrary high intensity. Consequently, we derive a semiclassical expression for the width of the LDOS which is shown to be very accurate for paradigmatic systems of quantum chaos. This Letter demonstrates the universal response of quantum systems with classically chaotic dynamics.

DOI
10.1103/physrevlett.104.254101

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The prediction of the response of a closed system to external perturbations is one of the central problems in quantum mechanics, and in this respect, the local density of states (LDOS) provides an in-depth description of such a response. The LDOS is the distribution of the overlaps squared connecting the set of eigenfunctions with the perturbed one. Here, we show that in the case of closed systems with classically chaotic dynamics, the LDOS is a Breit-Wigner distribution under very general perturbations of arbitrary high intensity. Consequently, we derive a semiclassical expression for the width of the LDOS which is shown to be very accurate for paradigmatic systems of quantum chaos. This Letter demonstrates the universal response of quantum systems with classically chaotic dynamics.
We study the quantum-classical correspondence for systems with interacting spin particles that are strongly chaotic in the classical limit. This is done in the presence of constants of motion associated with the fixed angular momenta of individual spins. Our analysis of the Lyapunov spectra reveals that the largest Lyapunov exponent agrees with the Lyapunov exponent that determines the local instability of each individual spin moving under the influence of all other spins. Within this picture, we introduce a rigorous and simple test of ergodicity for the spin motion, and use it to identify when classical chaos is both strong and global in phase space. In the quantum domain, our analysis of the Hamiltonian matrix in a proper representation allows us to obtain the conditions for the onset of quantum chaos as a function of the model parameters. From the comparison between the quantum and classical domains, we demonstrate that quantum quantities, such as the local density of states (LDoS) and the shape of the chaotic eigenfunctions written in the noninteracting many-body basis, have well-defined classical counterparts. We also find a relationship between the Kolmogorov-Sinai entropy and the width of the LDoS, which is useful for studies of many-body dynamics.
We compare the influence of perturbations of classically regular respectively chaotic Hamiltonians on their eigenfunctions. A generic measure of the perturbation strength is given by the average spreading width, which semiclassically is a phase-space integral. Thus, contrary to common assumption, the spreading width cannot be an indicator of regularity or chaos. The distribution of expansion coefficients of perturbed eigenfunctions in terms of unperturbed ones is markedly different for the two cases and may serve as a quantum signature of chaos referring to states rather than to spectra.
Even as we understand for long that the world is quantal and buried in it is classical dynamics which is chaotic, finding eigenfunctions analytically from the the Schroedinger equation has turned out to be a near-impossibility. Here, we discover a class of chaotic quantum systems for which we obtain some analytically exact eigenfunctions in closed form. This paves way to an exact classical and quantum mechanical treatment of chaotic systems. Furthermore, we bring out connections, underlying the discovery, between different areas of physics and mathematics related to universality in fluctuations observed in a wide variety of complex quantum systems.
We study the model of two interacting particles moving in a 1D box, paying main attention to the quantum-classical correspondence for the average shape of quantum eigenstates and for the local density of states (LDOS). We show that if the classical motion is chaotic, in a deep semi-classical region of a quantum system, both the shape of eigenstates and of the LDOS coincide with their classical analogs, on average. However, individual eigenstates exhibit quite large fluctuations which may not be treated as statistical ones. Thus, comparison of quantum quantities to the classical ones allows one to detect quantum effects of localization which for conservative systems emerge in the energy space.
Quantized, compact graphs are excellent paradigms for quantum chaos in bounded systems. Connecting them with leads to infinity, we show that they display all the features which characterize quantum chaotic scattering. We derive exact expressions for the scattering matrix, and an exact trace formula for the density of resonances, in terms of classical orbits, analogous to the semiclassical theory of chaotic scattering. A statistical analysis of the cross sections and resonance parameters compares well with the predictions of random matrix theory. Hence, this system is proposed as a convenient tool to study the generic behavior of chaotic scattering systems and their semiclassical description.
The average shape of the spectral local density of states (LDOS) and eigenfunctions (EFs) has been studied numerically for a conservative dynamical model (three-orbital Lipkin-Meshkov-Glick model) that can exhibit strong chaos in the classical limit. Attention is paid to the comparison of the shape of the LDOS with that known for random matrix models, as well as to the shape of the EFs, for different values of the perturbation strength. The classical counterparts of the LDOS has also been studied and found to be in remarkable agreement with the quantum calculations. Finally, by making use of a generalization of Brillouin-Wigner perturbation expansion, the form of the long tails of the LDOS and EFs is given analytically and confirmed numerically.