  The existence of frictionless flow below a critical velocity for obstacles
moving in a superfluid is well established in the context of the mean-field
Gross-Pitaevskii theory. We calculate the next order correction due to quantum
and thermal fluctuations and find a non-zero force acting on a delta-function
impurity moving through a quasi-one-dimensional Bose-Einstein condensate at all
subcritical velocities and at all temperatures. The force occurs due to an
imbalance in the Doppler shifts of reflected quantum fluctuations from either
side of the impurity. Our calculation is based on a consistent extension of
Bogoliubov theory to second order in the interaction strength, and finds new
analytical solutions to the Bogoliubov-de Gennes equations for a gray soliton.
Our results raise questions regarding the quantum dynamics in the formation of
persistent currents in superfluids.
