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arXiv:0711.4970 · Physical Review E

Scarring in open quantum systems

Diego A. Wisniacki, Gabriel G. Carlo

Published Apr 15, 2008• 31 citations
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Abstract

We study scarring phenomena in open quantum systems. We show numerical evidence that individual resonance eigenstates of an open quantum system present localization around unstable short periodic orbits in a similar way as their closed counterparts. The structure of eigenfunctions around these classical objects is not destroyed by the opening. This is exposed in a paradigmatic system of quantum chaos, the cat map.

DOI
10.1103/physreve.77.045201

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We study scarring phenomena in open quantum systems. We show numerical evidence that individual resonance eigenstates of an open quantum system present localization around unstable short periodic orbits in a similar way as their closed counterparts. The structure of eigenfunctions around these classical objects is not destroyed by the opening. This is exposed in a paradigmatic system of quantum chaos, the cat map.
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We introduce a new phase space representation for open quantum systems. This is a very powerful tool to help advance in the study of the morphology of their eigenstates. We apply it to two different versions of a paradigmatic model, the baker map. This allows us to show that the long-lived resonances are strongly scarred along the shortest periodic orbits that belong to the classical repeller. Moreover, the shape of the short-lived eigenstates is also analyzed. Finally, we apply an antiunitary symmetry measure to the resonances that allows us to quantify their localization on the repeller.
Chaos makes isolated systems of many interacting particles quickly thermalize and forget about their past. Here, we show that quantum mechanics hinders chaos in many-body systems: although the quantum eigenstates are thermal and strongly entangled, exponentially many of them are scarred, that is, have an enlarged weight along underlying classical unstable periodic orbits. Scarring makes the system more likely to be found on an orbit it was initialized on, retaining a memory of its past and thus weakly breaking ergodicity, even at long times and despite the system being fully thermal and the eigenstate thermalization hypothesis fulfilled. We demonstrate the ubiquity of quantum scarring in many-body systems by considering a large family of spin models, including some of the most popular ones from condensed matter physics. Our findings, at hand for modern quantum simulators, prove structure in spite of chaos in many-body quantum systems.
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.
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.
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.