  We realise a feedback-controlled optical Fabry-Perot cavity in which the
transmitted cavity output is used to modulate the input amplitude fluctuations.
The resulting phase-dependent fluctuations of the in-loop optical field, which
may be either sub-shot- or super-shot-noise, can be engineered to favorably
affect the optomechanical interaction with a nanomechanical membrane placed
within the cavity. Here we show that in the super-shot-noise regime
("anti-squashed light") the in-loop field has a strongly reduced effective
cavity linewidth, corresponding to an increased optomechanical cooperativity.
In this regime feedback improves the simultaneous resolved sideband cooling of
two nearly degenerate membrane mechanical modes by one order of magnitude.
