  It is well known that in the presence of a ring cavity the light scattering
from a uniform atomic ensemble can become unstable resulting in the collective
atomic recoil lasing. This is the result of a positive feedback due to the
cavity. We propose to add an additional electronic feedback loop based on the
photodetection of the scattered light. The advantage is a great flexibility in
choosing the feedback algorithm, since manipulations with electric signals are
very well developed. In this paper we address the application of such a
feedback to atoms in the Bose-Einstein condensed state and explore the quantum
noise due to the incoherent feedback action. We show that although the feedback
based on the photodetection does not change the local stability of the initial
uniform distribution with respect to small disturbances, it reduces the region
of attraction of the uniform equilibrium. The feedback-induced nonlinearity
enables quantum fluctuations to bring the system out of the stability region
and cause an exponential growth even if the uniform state is globally stable
without the feedback. Using numerical solution of the feedback master equation
we show that there is no feedback-induced noise in the quadratures of the
excited atomic and light modes. The feedback loop, however, introduces
additional noise into the number of quanta of these modes. Importantly, the
feedback opens an opportunity to position the modulated BEC inside a cavity as
well as tune the phase of scattered light. This can find applications in
precision measurements and quantum simulations.
