  In dissipative quantum systems, strong symmetries can lead to the existence
of conservation laws and multiple steady states. The investigation of such
strong symmetries and their consequences on the dynamics of the dissipative
systems is still in its infancy. In this work we investigate a strong symmetry
for bosonic atoms coupled to an optical cavity, an experimentally relevant
system, using adiabatic elimination techniques and numerically exact matrix
product state methods. We show the existence of multiple steady states for
ideal bosons coupled to the cavity. We find that the introduction of a weak
breaking of the strong symmetry by a small interaction term leads to a direct
transition from multiple steady states to a unique steady state. We point out
the phenomenon of dissipative freezing, the breaking of the conservation law at
the level of individual realizations in the presence of the strong symmetry.
For a weak breaking of the strong symmetry we see that the behavior of the
individual trajectories still shows some signs of this dissipative freezing
before it fades out for a larger symmetry breaking terms.
