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arXiv:0811.3017 · Physical Review Letters

Time-Domain Scars: Resolving the Spectral Form Factor in Phase Space

Thomas Dittrich, Leonardo A. Pachón

Published Apr 15, 2009• 12 citations
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Abstract

We study the relationship of the spectral form factor with quantum as well as classical probabilities to return. Defining a quantum return probability in phase space as a trace over the propagator of the Wigner function allows us to identify and resolve manifolds in phase space that contribute to the form factor. They can be associated with classical invariant manifolds such as periodic orbits, but also to nonclassical structures such as sets of midpoints between periodic points. In contrast to scars in wave functions, these features are not subject to the uncertainty relation and therefore need not show any smearing. They constitute important exceptions from a continuous convergence in the classical limit of the Wigner towards the Liouville propagator. We support our theory with numerical results for the quantum cat map and the harmonically driven quartic oscillator.

DOI
10.1103/physrevlett.102.150401

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We study the relationship of the spectral form factor with quantum as well as classical probabilities to return. Defining a quantum return probability in phase space as a trace over the propagator of the Wigner function allows us to identify and resolve manifolds in phase space that contribute to the form factor. They can be associated with classical invariant manifolds such as periodic orbits, but also to nonclassical structures such as sets of midpoints between periodic points. In contrast to scars in wave functions, these features are not subject to the uncertainty relation and therefore need not show any smearing. They constitute important exceptions from a continuous convergence in the classical limit of the Wigner towards the Liouville propagator. We support our theory with numerical results for the quantum cat map and the harmonically driven quartic oscillator.
We study nonergodic behavior of an ensemble of quantum cat map with initially localized phase-space distributions. For this purpose, an amended version of the Wigner function on discretized phase torus is given, which shows meaningful classical-quantum correspondence in the semiclassical limit. We find the time averaged coarse-grained Wigner function is scarred and antiscared along some short periodic orbits, and the heights (depths) of these scars (antiscars) decrease linearly with when the initial phase-space distribution is fixed and the semiclassical limit is taken by letting with a prime and .
Correct use of the Wigner representation of quantum mechanics, which is realized with joint distributions of quasiprobabilities in phase space, requires the use of certain specific rules and attention to a number of properties which distinguish the quasiprobability from a true probability. This paper is mainly concerned with these problems. In the Wigner representation the quantum Liouville equation appears instead of the Schroedinger equation. The solution may have no physical meaning unless it is subjected to a necessary and sufficient condition which selects an allowed class of distributions which describe quantum-mechanical pure states. This condition contains Planck's constant and imposes, besides the uncertainty relations, severe restrictions on the possible form of the Wigner function. When this condition is satisfied, one can reconstruct the wave function from the Wigner function. In the case of an oscillator, the quantization condition for the energies of the stationary states does not follow from the Liouville equation, but from this supplementary condition. Unlike the true probability density, any Wigner function (except Gaussian ones) of a pure state takes on negative values. Another important peculiarity is that the quasiprobability is not concentrated on certain hypersurfaces, but is smeared out over the entire phase space. These and other features considered in this paper should be kept in mind when using the Wigner representation in quantum-mechanical problems.
We propose a picture of Wigner function scars as a sequence of concentric rings along a two-dimensional surface inside a periodic orbit. This is verified for a two-dimensional plane that contains a classical hyperbolic orbit of a Hamiltonian system with 2 degrees of freedom. The stationary wave functions are the familiar mixture of scarred and random waves, but the spectral average of the Wigner functions in part of the plane is nearly that of a harmonic oscillator and individual states are also remarkably regular. These results are interpreted in terms of the semiclassical picture of chords and centers.
We present here a study of the quantum phase space localization (Wigner functions) in dependence of semiclassical quantization rules for the hydrogen atom in magnetic fields. We consider primarily two energy regions. In the first the corresponding classical system is near-integrable and in the second near fully chaotic. We study phase space localization (scars) close to stable and unstable trajectories, tori and the invariant manifold associated with the almost circular, unstable periodic trajectory. We find that these classical structures are an important element for the structure of the wavefunctions and the semiclassical quantization predictions are in good agreement with the quantal results.
Quantum trajectories defined in the de Broglie-Bohm theory provide a causal way to interpret physical phenomena. In this letter, we use this formalism to analyze the short-time dynamics induced by unstable periodic orbits in a classically chaotic system, a situation in which scars are known to play a very important role. We find that the topologies of the quantum orbits are much more complicated than that of the scarring and associated periodic orbits, since the former have quantum interference built in. Thus scar wave functions are necessary to analyze the corresponding dynamics. Moreover, these topologies imply different return routes to the vicinity of the initial positions, and this reflects in the existence of different contributions in each peak of the survival probability function.
Unstable periodic orbits (UPOs) play a key role in the theory of chaos, constituting the ``skeleton'' of classical chaotic systems and ``scarring'' the eigenstates of the corresponding quantum system. Recently, nonthermal many-body eigenstates embedded in an otherwise thermal spectrum have been identified as a many-body generalization of quantum scars. The latter, however, are not clearly associated to a chaotic phase space, and the connection between the single- and many-body notions of quantum scars remains therefore incomplete. Here, we find the first quantum many-body scars originating from UPOs of a chaotic phase space. Remarkably, these states verify the eigenstate thermalization hypothesis, and we thus refer to them as thermal quantum many-body scars. While they do not preclude thermalization, their spectral structure featuring approximately equispaced towers of states yields an anomalous oscillatory dynamics preceding thermalization for wavepackets initialized on an UPO. Remarkably, our model hosts both types of scars, thermal and nonthermal, and allows us to study the crossover between the two. Our work illustrates the fundamental principle of classical-quantum correspondence in a many-body system and its limitations.