  We use the Bethe Ansatz technique to study dissipative systems experiencing
loss. The method allows us to exactly calculate the Liouvillian spectrum. This
opens the possibility of analytically calculating the dynamics of a wide range
of experimentally relevant models including cold atoms subjected to one and two
body losses, coupled cavity arrays with bosons escaping the cavity, and cavity
quantum electrodynamics. As an example of our approach we study the relaxation
properties in a boundary driven XXZ spin chain. We exactly calculate the
Liouvillian gap and find different relaxation rates with a novel type of
dynamical dissipative phase transition. This physically translates into the
formation of a stable domain wall in the easy-axis regime despite the presence
of loss. Such analytic results have previously been inaccessible for systems of
this type.
