  We theoretically investigate the quasiparticle (polaron) properties of an
impurity immersing in a Bose-Einstein condensate with equal Rashba and
Dresselhaus spin-orbit coupling at zero temperature. In the presence of
spin-orbit coupling, all bosons can condense into a single plane-wave state
with finite momentum, and the corresponding excitation spectrum shows an
intriguing roton minimum. We find that the polaron properties are strongly
modified by this roton minimum, where the ground state of attractive polaron
acquires a nonzero momentum and anisotropic effective mass. Across the
resonance of the interaction between impurity and atoms, the polaron evolves
into a tight-binding dimer. We show that the evolution is not smooth when the
roton structure of the condensate becomes apparent, and a first-order phase
transition from a phonon-induced polaron to a roton-induced polaron is observed
at a critical interaction strength.
