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Acta Physica Polonica A

Quantum Graphology

Tsampikos Kottos

Published Jan 1, 2006• 2 citations
Publisher

Abstract

We review quantum chaos on graphs.We construct a unitary operator which represents the quantum evolution on the graph and study its spectral and wave function statistics.This operator is the analogue of the classical evolution operator on the graph.It allows us to establish a connection between the corresponding periodic orbits and the statistical properties of eigenvalues and eigenfunctions.Specifically, for the energy-averaged spectral form factor we derived an exact combinatorial expression which illustrate the role of correlations between families of isometric orbits.We also show that enhanced wave function localization due to the presence of short unstable periodic orbits and strong scarring can rely on completely different mechanisms.

DOI
10.12693/aphyspola.109.7

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We review quantum chaos on graphs.We construct a unitary operator which represents the quantum evolution on the graph and study its spectral and wave function statistics.This operator is the analogue of the classical evolution operator on the graph.It allows us to establish a connection between the corresponding periodic orbits and the statistical properties of eigenvalues and eigenfunctions.Specifically, for the energy-averaged spectral form factor we derived an exact combinatorial expression which illustrate the role of correlations between families of isometric orbits.We also show that enhanced wave function localization due to the presence of short unstable periodic orbits and strong scarring can rely on completely different mechanisms.
We review quantum chaos on graphs. We construct a unitary operator which represents the quantum evolution on the graph and study its spectral and wavefunction statistics. This operator is the analogue of the classical evolution operator on the graph. It allow us to establish a connection between the corresponding periodic orbits and the statistical properties of eigenvalues and eigenfunctions. Specifically, for the energy-averaged spectral form factor we derived an exact combinatorial expression which illustrate the role of correlations between families of isometric orbits. We also show that enhanced wave function localization due to the presence of short unstable periodic orbits and strong scarring can rely on completely different mechanisms.
We review recent progress in attaining a quantitative understanding of the scarring phenomenon, the non-random behaviour of quantum wavefunctions near unstable periodic orbits of a classically chaotic system. The wavepacket dynamics framework leads to predictions about statistical long-time and stationary properties of quantum systems with chaotic classical analogues. Many long-time quantum properties can be quantitatively understood using only short-time classical dynamics information; these include wavefunction intensity distributions, intensity correlations in phase space and correlations between wavefunctions, and distributions of decay rates and conductance peaks in weakly open systems. Strong deviations from random matrix theory are predicted and observed in the presence of short unstable periodic orbits.
We investigate the role of quantum monitoring in the dynamical manifestations of Hamiltonian quantum chaos. Specifically, we analyze the generalized spectral form factor, defined as the survival probability of a coherent Gibbs state under continuous energy measurements. We show that quantum monitoring can tailor the signatures of quantum chaos in the dynamics, such as the extension of the ramp in the spectral form factor, by varying the measurement strength and detection efficiency. In particular, a typical quantum trajectory obtained by monitoring with unit efficiency exhibits enhanced quantum chaos relative to the average dynamics and to unitary evolution without measurements. Our results hold for systems across symmetry classes and for other diagnostics of quantum chaos, such as out-of-time-order correlators and Liouvillian spectral statistics.
We investigate the appearance of quantum chaos in a single many-body wave function by analyzing the statistical properties of the eigenvalues of its reduced density matrix ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{A}$ of a spatial subsystem AAA. We find that (i): the spectrum of the density matrix is described by so-called Wishart random matrix theory, which (ii): exhibits besides level repulsion, spectral rigidity, and universal spectral correlations between eigenvalues separated by distances ranging from one up to many mean level spacings, which we investigate. We use these universal spectral characteristics of the reduced density matrix as a definition of chaos in the wave function. A simple and precise characterization of such universal correlations in a spectrum is a segment of strictly linear growth at sufficiently long times, recently called the ``ramp,'' of the spectral form factor which is the Fourier transform of the correlation function between a pair of eigenvalues. It turns out that Wishart and standard random matrix theory have the same universal ``ramp.'' Specifically, here numerical results for the spectral form factor of the density matrix of generic nonintegrable many-body systems, such as one-dimensional quantum Ising and Floquet spin models, are found to exhibit a universal ``ramp'' identical to that appearing for a ``random pure state'' (``Page state,'' or ``Haar state''). The density matrix of the latter is precisely the Wishart random matrix, the reduced density matrix of a completely random wave function. In addition, we study the development of chaos in the wave function by letting an initial direct product state evolve under the unitary time evolution. We find that the universal spectral correlations as manifested by the ``ramp'' set in as soon as the entanglement entropy begins to grow, and first develop for the eigenvalues at the top of the spectrum of the density matrix ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{A}$, subsequently spreading over the entire spectrum at later times. Finally, we study a prethermalized regime described by a generalized Gibbs ensemble, which develops in a rapidly driven Floquet model at intermediate times. We find that the prethermalized regime exhibits no chaos, as evidenced by the absence of a ``ramp'' in the spectral form factor of the density matrix, while the universal spectral correlations start to develop when the prethermalized regime finally relaxes at late times to the fully thermalized (infinite temperature) chaotic regime.