  This paper extends previous findings on the modified Schr\"odinger evolution
inspired by quantum gravity phenomenology. By establishing a connection between
this approach and fractional quantum mechanics, we provide insights into a
potential deep infrared regime of quantum gravity, characterized by the
emergence of fractal dimensions, similar to behaviors observed in the deep
ultraviolet regime. Additionally, we briefly explore the requirements for
experimental investigations of this regime using Bose-Einstein condensates.
Notably, our analysis reveals a direct implication of this analogy: experiments
probing fractional quantum mechanics may serve as equivalent models of quantum
gravity. We identify instances of nonlocal behavior in such systems, suggesting
an analogous phenomenon of nonlocality in quantum gravity.
