Estimating ergodization time of a chaotic many-particle system from a time reversal of equilibrium noise

Abstract

We propose a method of estimating ergodization time of a chaotic many-particle system by monitoring equilibrium noise before and after time reversal of dynamics(Loschmidt echo). The ergodization time is defined as the characteristic time required to extract the largest Lyapunov exponent from a system's dynamics. We validate the method by numerical simulation of an array of coupled Bose-Einstein condensates in the regime describable by the discrete Gross-Pitaevskii equation. The quantity of interest for the method is a counterpart of out-of-time-order correlators in the quantum regime. Outline of the methodIn figure 1, we outline the method. It consists of the following steps.

New characteristic of quantum many-body chaotic systems

“…We provide analytic and numerical evidence that γ 0 is at most polynomially dependent on L −1 . It has been shown recently that γ (x), which has the dimension of energy, is related to the characteristic timescale of thermalization t 0 [21], thus connecting chaos with thermalization dynamics [9,14,[22][23][24][25][26][27]. Finally, we note that E(x) provides an efficient way to distinguish chaotic systems from the nonchaotic ones.…”

Quantum and Classical Lyapunov Exponents in Atom-Field Interaction Systems

“…For interacting many-body systems, while exponential behaviors for the OTOC have been found for the Sachdev-Ye-Kitaev model [11,22] and for the Bose-Hubbard model [23,24], a direct demonstration of the quantum-classical correspondence has not yet been made. Studies in this direction include [6,[25][26][27][28][29] and [30].Here, we investigate the OTOC for the Dicke model [31,32]. Comparing with one-body systems, the model is a step up toward an explicit quantum-classical correspondence for interacting many-body systems, since it contains N atoms interacting with a quantized field.The Dicke model was originally proposed to explain the collective phenomenon of superradiance: the field mediates interatomic interactions, which causes the atoms to act collectively [31,33].…”

Quantum echo dynamics in the Sherrington-Kirkpatrick model

“…In the case of the transverse field h = 0.6 J as in Fig. 5, an exponential fit yields 2Λ 1.5/J, while in general Λ is an increasing function of h. The rate Λ (sometimes referred to as the generalized Lyapunov exponent [76][77][78]) is related to the maximal Lyapunov exponent of the theory λ ma x [24,29]. The difference between the two comes from the different order of operations of taking logarithm and ensemble averaging.…”