Abstract
The analysis of level statistics provides a primary method to detect signatures of chaos in the quantum domain. However, for experiments with ion traps and cold atoms, the energy levels are not as easily accessible as the dynamics. In this work, we discuss how properties of the spectrum that are usually associated with chaos can be directly detected from the evolution of the number operator in the one-dimensional, noninteracting Aubry-André model. Both the quantity and the model are studied in experiments with cold atoms. We consider a single-particle and system sizes experimentally reachable. By varying the disorder strength within values below the critical point of the model, level statistics similar to those found in random matrix theory are obtained. Dynamically, these properties of the spectrum are manifested in the form of a dip below the equilibration point of the number operator. This feature emerges at times that are experimentally accessible. This work is a contribution to a special issue dedicated to Shmuel Fishman.
Highlights
There has been a surprising revival of interest in quantum chaos, especially from a dynamical perspective, with the exponential growth of out-of-time ordered correlators (OTOC) taken as a main indication of chaotic behavior [1,2,3,4,5,6,7,8,9]
We propose a way to directly detect the effects of level repulsion in the evolution of a quantum system
This work shows that the effects of level repulsion can be directly observed by studying the evolution of the number operator in the finite one-particle 1D Aubry-André model
Summary
There has been a surprising revival of interest in quantum chaos, especially from a dynamical perspective, with the exponential growth of out-of-time ordered correlators (OTOC) taken as a main indication of chaotic behavior [1,2,3,4,5,6,7,8,9]. All states in the Aubry-André model become localized only above a critical disorder strength, while in the one-particle 1D infinite Anderson model, all states are localized for any disorder strength [28,29,30,31] Despite this difference, when the systems are finite and have small disorder strengths, they present similar level spacing distributions; namely, they show distributions as in RMT, the so-called Wigner–Dyson distributions [32,33]. Two other problems remain: the correlation hole in systems with many particles emerges at extremely low values of the survival probability, and this quantity is non-local in real space, while experiments usually deal with local quantities (exceptions include [46]) To solve these problems, we consider the one-particle 1D Aubry-André model and study the evolution of the number operator. We expect our results to be valid in this broader picture
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