Glimmers of a Quantum KAM Theorem: Insights from Quantum Quenches in One Dimensional Bose Gases

TL;DR

This paper explores how weak integrability breaking in one-dimensional Bose gases leads to residual quasi-conserved quantities, offering a quantum analog to the classical KAM theorem and providing insights into quantum dynamics and thermalization.

Contribution

It introduces the concept of residual quasi-conserved quantities in weakly non-integrable quantum systems, extending the classical KAM theorem to the quantum domain.

Findings

Construction of residual quasi-conserved quantities in quantum systems.

Demonstration of these quantities in quantum quenches of 1D Bose gases.

Potential experimental probes of the quasi-conserved quantities.

Abstract

Real-time dynamics in a quantum many-body system are inherently complicated and hence difficult to predict. There are, however, a special set of systems where these dynamics are theoretically tractable: integrable models. Such models possess non-trivial conserved quantities beyond energy and momentum. These quantities are believed to control dynamics and thermalization in low dimensional atomic gases as well as in quantum spin chains. But what happens when the special symmetries leading to the existence of the extra conserved quantities are broken? Is there any memory of the quantities if the breaking is weak? Here, in the presence of weak integrability breaking, we show that it is possible to construct residual quasi-conserved quantities, so providing a quantum analog to the KAM theorem and its attendant Nekhoreshev estimates. We demonstrate this construction explicitly in the context of quantum quenches in one-dimensional Bose gases and argue that these quasi-conserved quantities can be probed experimentally.