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 nontrivial 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 quasiconserved quantities, thus 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 quasiconserved quantities can be probed experimentally.
6 More- Received 30 August 2014
DOI:https://doi.org/10.1103/PhysRevX.5.041043
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Published by the American Physical Society
Popular Summary
The Kolmogorov-Arnold-Moser (KAM) theorem is a famed theorem in classical mechanics governing the crossover from integrability to chaos in the dynamics of a classical system. According to the KAM theorem, chaos is not ubiquitous; special classical systems exist with a number of exotic conserved quantities (beyond energy and momentum) known as integrable systems. The motion of bodies in an integrable system appears to be nonchaotic. However, an open question is how generic integrable systems are. In nature, no system is ever truly integrable; there are always small corrections that spoil integrability. The KAM theorem states that there is a smooth crossover between integrable and chaotic dynamics; breaking the special symmetries of an integrable model does not immediately lead to chaotic behavior.
Here, we provide a direct quantum analog to the KAM theorem. Using a one-dimensional Bose gas in a one-body parabolic trap that is then released suddenly into a small, one-body cosine potential, we demonstrate a direct quantum variant of the KAM theorem. Absent the one-body potential, the gas is described by the Lieb-Liniger model, an integrable model. In the presence of the cosine potential, however, integrability is broken. We show that the conserved quantities of the integrable Lieb-Liniger model are not immediately destroyed in the presence of the integrability breaking cosine. Rather, the model’s conserved quantities are deformed and, at least on the low-energy Hilbert space, remain nearly conserved. This deformation amounts to taking linear combinations of the original charges.
We demonstrate that by increasing the number of charges in the linear combination, we can control the quality of the quasiconservation.