  We consider a Bose-Einstein condensate interacting with a gravitational wave
for the case when the gravitational fluctuations are quantized in order to
incorporate quantum gravity effects into the theory. We observe that the
solution of the time-dependent part of the pseudo-Goldstone boson has infusions
from the noise induced by gravitons and the corresponding differential equation
of motion is Langevin-like. Using this result, we obtain the quantum gravity
modified Fisher information which has been termed as the quantum gravitational
Fisher information (QGFI). The inverse square root of the stochastic average of
the QGFI gives the minimum uncertainty in the measurement of the gravitational
wave amplitude. The minimum uncertainty does not go to infinity as the
measurement time approaches zero in a quantum gravity setup rather it has a
measurable finite value for gravitons with high squeezing. Finally, we observe
the effect of decoherence due to interacting phonon modes in the QGFI and
observe a less obvious decoherence effect for higher squeezing of the initial
graviton.
