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Communications in Theoretical Physics

Influence of Scars on Time Averaged Ensemble Distribution of Quantum Cat Map in the Semiclassical Regime

Jiao Wang, Yan Gu

Published Oct 30, 1999• 0 citations
Publisher

Abstract

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 .

DOI
10.1088/0253-6102/32/3/361

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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 .
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 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 predict and numerically observe strong periodic orbit effects in the properties of weakly open quantum systems with a chaotic classical limit. Antiscars lead to a large number of exponentially narrow isolated resonances when the single-channel (or tunneling) opening is located on a short unstable orbit of the closed system; the probability to remain in the system at long times is thus exponentially enhanced over the random matrix theory prediction. The distribution of resonance widths and the probability to remain are quantitatively given in terms of only the stability matrix of the orbit on which the opening is placed. The long-time remaining probability density is nontrivially distributed over the available phase space; it can be enhanced or suppressed near orbits other than the one on which the lead is located, depending on the periods and classical actions of these other orbits. These effects of the short periodic orbits on quantum decay rates have no classical counterpart, and first appear on times scales much larger than the Heisenberg time of the system. All the predictions are quantitatively compared with numerical data.
Chaos plays a crucial role in numerous natural phenomena, but its quantum nature has remained large elusive. One intriguing quantum-chaotic phenomenon is the scarring of a single-particle wavefunction, where the quantum probability density is enhanced in the vicinity of a classical periodic orbit. These quantum scars illustrate the quantum suppression of classical chaos, offering a unique way to explore the classical-quantum relationship beyond conventional limits. In this study, we establish an ergodicity theorem for slacking a group of adjacent eigenstates, revealing the aspect of antiscarring -- the reduction of probability density along a periodic orbit generating the corresponding scars. We thereafter apply these two concepts to variational scars in a disordered quantum well, and finally discuss their broader implications, suggesting potential experimental approaches to observe this phenomenon.
We study the stability of classical structures in chaotic systems when a dissipative quantum evolution takes place. We consider a paradigmatic model, the quantum baker map in contact with a heat bath at finite temperature. We analyze the behavior of the purity, fidelity and Husimi distributions corresponding to initial states localized on short periodic orbits (scar functions) and map eigenstates. Scar functions, that have a fundamental role in the semiclassical description of chaotic systems, emerge as robust relative to other states (which are localized on classical structures) against environmental perturbations. Also, purity and fidelity show a complementary behavior as decoherence measures.
Quantum many-body scars (QMBSs) represent a weak ergodicity-breaking phenomenon that defies the common scenario of thermalization in closed quantum systems. They are often regarded as a many-body analog of quantum scars (QSs)—a single-particle phenomenon in quantum chaos—due to their superficial similarities. However, unlike QSs, a clear connection between QMBSs and classical chaos has remained elusive. It has nevertheless been speculated that in an appropriate semiclassical limit, QMBSs should have a correspondence to weakly unstable periodic orbits. In this paper, I present a counterexample to this conjecture by studying a bosonic model with a large number of flavors. The dynamics of out-of-time-ordered correlators (OTOCs) suggest that QMBS do not display chaotic behavior in the semiclassical limit. In contrast, chaotic dynamics are expected for initial states not associated with QMBS. Interestingly, the anomalous OTOC dynamics persist even under weak perturbations that eliminate the scarred eigenstates, suggesting a certain robustness in the phenomenon.