Out-of-time-order correlators in quantum mechanics (1703.09435v1)

Abstract: The out-of-time-order correlator (OTOC) is considered as a measure of quantum chaos. We formulate how to calculate the OTOC for quantum mechanics with a general Hamiltonian. We demonstrate explicit calculations of OTOCs for a harmonic oscillator, a particle in a one-dimensional box, a circle billiard and stadium billiards. For the first two cases, OTOCs are periodic in time because of their commensurable energy spectra. For the circle and stadium billiards, they are not recursive but saturate to constant values which are linear in temperature. Although the stadium billiard is a typical example of the classical chaos, an expected exponential growth of the OTOC is not found. We also discuss the classical limit of the OTOC. Analysis of a time evolution of a wavepacket in a box shows that the OTOC can deviate from its classical value at a time much earlier than the Ehrenfest time.

Overview of the Paper "Out-of-time-order correlators in quantum mechanics"

This paper presents a thorough examination of Out-of-Time-Order Correlators (OTOCs) within the framework of quantum mechanics, exploring their characteristics as indicators of quantum chaos across differing quantum systems. The authors, Koji Hashimoto, Keiju Murata, and Ryosuke Yoshii, propose a rigorous method to calculate OTOCs for systems governed by a general Hamiltonian, a task executed through comprehensive numerical investigations on specific quantum systems.

Methodology and Systems Analyzed

The paper begins with an articulation of the OTOC definition, with a focus on its utility as a metric of chaos, both in quantum and classical contexts. The authors delve into the practical computation of OTOCs using quantum mechanical systems with explicitly defined Hamiltonians. Notable examples explored include:

1. Harmonic Oscillator: Here, the OTOCs exhibit periodic behavior reflective of the regular, harmonic nature of the system's energy spectrum.
2. Particle in a One-Dimensional Box: The paper addresses this well-known model's commensurable energy spectrum, yielding periodic OTOC outcomes.
3. Circle Billiard: While comparable in geometry to classical systems, the quantum mechanical approach elucidates saturation to constants at high temperatures, diverging markedly from classical chaos expectancies.
4. Stadium Billiard: A paradigmatic case of classical chaos, yet the anticipated exponential OTOC growth aligning with chaos is notably absent in findings.

Key Findings

• Growth and Saturation: The results identify that OTOCs show initial growth phases but culminate in saturation, marked by temperature-dependent constant values. This saturation contrasts distinctly with the prolonged exponential growth predicted by classical chaotic mechanics. For instance, in stadium billiards, while classical mechanics suggests high sensitivity to initial conditions (chaos), the quantum results do not reflect this in the OTOC growth rates.
• Absence of Expected Quantum Lyapunov Exponents: Particularly in the case of stadium billiards, surprising results are observed. Despite classical indications of chaotic behavior, the associated relaxation to constant OTOC values with no notable exponential growth hints at a disjoint between quantum and classical chaos metrics in this context.
• Long-term Behavior and Temperature Dependence: The analyses uncovered an empirical constant, $C_T \sim m T \times (\text{system size})^2$, providing a straightforward linear relation linking OTOC magnitudes to temperature and system dimensions.

Implications and Speculations

The paper poses significant implications for the understanding of chaos in quantum systems. The absence of exponential OTOC growth, even in traditionally chaotic systems like the stadium billiard, challenges the classical understanding of chaos's translation to the quantum field. These findings bring into question whether the chaotic characteristics observed in classical mechanics can be universally applied to quantum systems, indicating a need for further theoretical development regarding quantum chaos theory.

The paper notably refrains from claiming a universal OTOC behavior across quantum systems, hinting at a complex bridge (or lack thereof) between quantum mechanics and classical chaos, augmented by features distinctive of quantum systems such as wave interference and the Heisenberg uncertainty relation's effects. The exploration of OTOCs across a broader class of quantum systems remains a fertile ground for future research.

These insights spur speculation on the potential for large-scale quantum systems (involving numerous interacting particles) to more visibly exhibit classical chaos attributes, potentially bridged by decoherence effects. Interdisciplinary advances, particularly within quantum gravity and AdS/CFT contexts, might further elucidate these complex dynamics.

In summary, while providing a solid computational framework and producing nuanced results, this paper emphasizes the enigmatic and often non-classical nature of quantum chaos, a domain ripe with opportunities for ongoing exploration and theoretical refinement.