Shadowself!
Welcome back.
The LIGO experiment as you quote above, at first glance, appears to be very very similar to the Michelson-Morley experiment of 1887.
It compared the speed of light in perpendicular directions, in an attempt to detect the relative motion of matter through the stationary luminiferous aether ("aether wind"). The result was negative, in that the expected difference between the speed of light in the direction of movement through the presumed aether, and the speed at right angles, was found not to exist; this result is generally considered to be the first strong evidence against the then-prevalent aether theory, and initiated a line of research that eventually led to special relativity, which rules out a stationary aether.
If memory serves, if the speed of light where altered by the movement of the medium it was thought to be propagating within, an interference pattern should emerge.
The results never did yield an interference pattern and so it was concluded that the theory of a medium of aether that light propagates through was invalid.
The text you quote above suggests an experiment of very similar setup, except where we have 4km of perpendicular legs... And we are trying to detect a difference within the width of an atom.
Do you feel up to commenting on this issue?