  We study the polaron problem of an impurity immersed in a dissipative
spin-orbit coupled Fermi gas via a non-self-consistent T-matrix method. We
first propose an experimental scheme to realize a spin-orbit coupled Fermi bath
with dissipation, and show that such a system can be described by a
non-Hermitian Hamiltonian that contains an imaginary spin-flip term and an
imaginary constant shift term. We find that the non-Hermiticity will change the
single particle dispersion of the bath gas, and modify the properties of
attractive and repulsive polarons such as energy, quasi-particle residue,
effective mass, and decay rate. We also investigate the Thouless criteria
corresponding to the instability of the polaron molecule transition, which
suggests a molecule state is more facilitated with stronger bath dissipation.
Finally, we consider the case with finite impurity density and calculate the
interaction between polarons. Our result extends the study of polaron physics
to non-Hermitian systems and may be realized in future experiment.
