Out-of-time-order correlators and quantum chaos

Abstract

Quantum Chaos has originally emerged as the field which studies how the properties of classical chaotic systems arise in their quantum counterparts. The growing interest in quantum many-body systems, with no obvious classical meaning has led to consider time-dependent quantities that can help to characterize and redefine Quantum Chaos. This article reviews the prominent role that the out of time ordered correlator (OTOC) plays to achieve such goal.

Relative asymptotic oscillations of the out-of-time-ordered correlator as a quantum chaos indicator

“…The time dependence of a general OTOC can be divided into short-time and long-time regimes, roughly limited by the Ehrenfest (or scrambling) time τ E ∝ λ −1 cl ln N [10,19,69,70], where N is the size parameter of the system. In more detailed analyses, there has also been described a universal power-law growth at very short times t d < t E [14] (t d is called the dephasing time) for OTOC operators commuting at t = 0, and yet another time scale given by the diffusion time t D t E , which is the time of complete saturation of the quantum dynamics [71].…”

“…They were introduced long ago as a semiclassical tool to study superconductivity [4] and later dusted off by showing their relevance in black-hole physics and chaos [5,6]. In the case of a quantum system with the classical limit, the short-time behavior of the OTOCs mimics the exponential spreading of neighboring classical trajectories up to the Ehrenfest time, leading to the notion of quantum Lyapunov exponent [7][8][9][10][11][12][13][14][15][16][17][18][19] and quantum butterfly effect [5,18,[20][21][22][23]. In quantum systems with local interactions and local OTOC operators, the initial time evolution of the OTOCs describes the spreading (or scrambling) of quantum information [24][25][26][27][28][29][30].…”

Chaos and multifold complexity for an inverted harmonic oscillator

“…To test the above conjecture, namely, the possible universality of the multifold complexity in such a limit, we will study a system with no classical holographic dual, which is nevertheless chaotic. For concreteness, we will examine the inverted harmonic oscillator, a system that has been widely studied in the past in terms of its OTOCs, the Loschmidt echo, and circuit complexity [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61]. The Loschmidt echo examines the inner product of bidirectional evolution.…”

Quantum chaos in the Dicke model and its variants

“…This scrambling can be measured using outof-time-ordered correlators (OTOCs), which was conjectured to have a bound on its growth rate [29][30][31]. The OTOC, first established in the context of superconductivity [32], is now presented as a measure of quantum chaos, with its growth rate attributed to the classical Lyapunov exponent [33][34][35][36][37][38]. Recently, OTOC has started receiving considerable attention because of its appearance as a valuable tool for generalizing quantum chaos studies beyond the time-independent, onebody case.…”