Integrability breaking in the one-dimensional Bose gas: Atomic losses and energy loss

The one-dimensional δ-function interacting Bose gas (the Lieb-Liniger model) is an integrable system, which can model experiments with ultra-cold atoms in one-dimensional traps. Even though the model is integrable, integrability breaking effects are always present in the real-world experiments. In this work we consider the integrability breaking due to atomic loss, which is the most relevant effect in the experiments. We set up a framework for the exact computation of the losses of the canonical charges of the model, and compute an exact result for the energy loss due to the local K-body processes, valid for arbitrary K. Our result takes the form of multiple integrals, which are explicitly factorized in the experimentally relevant cases of K=1,2,3.