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Does scarring prevent ergodicity

Saúl Pilatowsky-Cameo, David Villaseñor, M. A. Bastarrachea-Magnani, Sergio Lerma-Hernández, Léa Ferreira dos Santos, Jorge G. Hirsch

Published Sep 1, 2020• 3 citations
OpenAlex

Abstract

Classically chaotic systems are ergodic, that is after a long time, any trajectory will be arbitrarily close to any point of the available phase space, filling it uniformly. Using Born's rules to connect quantum states with probabilities, one might then expect that all chaotic quantum states should be uniformly distributed in phase space. This simplified picture was shaken by the discovery of quantum scarring, where some eigenstates are concentrated along unstable periodic orbits. Despite of that, it is consensus that most eigenstates of chaotic models are indeed ergodic. Our results show instead that all eigenstates of the chaotic Dicke model are actually scarred. Even the most random states of this interacting atom-photon system never occupy more than half of the available phase space. Quantum ergodicity is achieved only as an ensemble property, after temporal averages are performed covering the phase space.

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Classically chaotic systems are ergodic, that is after a long time, any trajectory will be arbitrarily close to any point of the available phase space, filling it uniformly. Using Born's rules to connect quantum states with probabilities, one might then expect that all chaotic quantum states should be uniformly distributed in phase space. This simplified picture was shaken by the discovery of quantum scarring, where some eigenstates are concentrated along unstable periodic orbits. Despite of that, it is consensus that most eigenstates of chaotic models are indeed ergodic. Our results show instead that all eigenstates of the chaotic Dicke model are actually scarred. Even the most random states of this interacting atom-photon system never occupy more than half of the available phase space. Quantum ergodicity is achieved only as an ensemble property, after temporal averages are performed covering the phase space.
Abstract In a classically chaotic system that is ergodic, any trajectory will be arbitrarily close to any point of the available phase space after a long time, filling it uniformly. Using Born’s rules to connect quantum states with probabilities, one might then expect that all quantum states in the chaotic regime should be uniformly distributed in phase space. This simplified picture was shaken by the discovery of quantum scarring, where some eigenstates are concentrated along unstable periodic orbits. Despite that, it is widely accepted that most eigenstates of chaotic models are indeed ergodic. Our results show instead that all eigenstates of the chaotic Dicke model are actually scarred. They also show that even the most random states of this interacting atom-photon system never occupy more than half of the available phase space. Quantum ergodicity is achievable only as an ensemble property, after temporal averages are performed.
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.
Unlike classical systems, understanding ergodicity from phase space mixing remains unclear for interacting quantum systems due to the absence of phase space trajectories. By considering an interacting spin model known as kicked coupled top, we elucidate the manifestation of phase space dynamics on local ergodic behavior of its quantum counterpart and quantum scarring phenomena. A transition to chaos occurs by increasing the kicking strength, and in the mixed phase space, the islands of regular motions within the chaotic sea clearly exhibit deviation from ergodicity, which we quantify from entanglement entropy and survival probability. Interestingly, the reminiscence of unstable orbits and fixed points can be identified as scars in quantum states, exhibiting athermal behavior and violation of Berry's conjecture for ergodic states. We also discuss the detection of quantum scars by a recently developed method of "out-of-time-order correlators," which has experimental relevance.
We reveal a feature of quantum scarring in systems with many particles: quantum scars, living densely near an unstable periodic orbit, must be compensated by corresponding antiscarred states suppressed there to establish the uniformity of the whole. The uniformity of the underlying phase space is linked to early-time dynamics---a regime beyond the predictions of random matrix theory and encapsulated in the eigenstate stacking theorem. By extending the domain of the stacking theorem, we apply our theory to a chaotic spinor Bose-Einstein condensate, whose quantum scar dynamics have recently been observed in the laboratory. Our work uncovers how scarring of some eigenstates affects the rest of the chaotic and thermal spectrum in quantum systems with many particles.
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.
Quantum many-body scars consist of a few low-entropy eigenstates in an otherwise chaotic many-body spectrum, and can weakly break ergodicity resulting in robust oscillatory dynamics. The notion of quantum many-body scars follows the original single-particle scars introduced within the context of quantum billiards, where scarring manifests in the form of a quantum eigenstate concentrating around an underlying classical unstable periodic orbit. A direct connection between these notions remains an outstanding problem. Here, we study a many-body spinor condensate that, owing to its collective interactions, is amenable to the diagnostics of scars. We characterize the system's rich dynamics, spectrum, and phase space, consisting of both regular and chaotic states. The former are low in entropy, violate the eigenstate thermalization hypothesis, and can be traced back to integrable effective Hamiltonians, whereas most of the latter are scarred by the underlying semiclassical unstable periodic orbits, while satisfying the eigenstate thermalization hypothesis. We outline an experimental proposal to probe our theory in trapped spin-1 Bose-Einstein condensates.