  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.
