  The effect of noise induced by gravitons has been investigated using a
Bose-Einstein condensate. The gravitational wave perturbation is then considerd
as a sum of discrete Fourier modes in the momentum space. Coming to an
operatorial representation and quantizing the phase space variables via
appropriately introduced canonincal commutation relations between the
canonically conjugate variables corresponding to the graviton and bosonic part
of the total system, one obtains a proper quantum gravity setup. Then we obtain
the Bogoliubov coefficients from the solution of the time-dependent part of the
pseudo-Goldstone boson and construct the covariance metric for the bosons
initially being in a squeezed state. Using the stochastic average of the Fisher
information, we obtain a lower bound on the amplitude parameter of the
gravitational wave. As the entire calculation is done at zero temperature, the
bosonic system, by construction, will behave as a Bose-Einstein condensate. For
a Bose-Einstein condensate with a single mode, we observe that the lower bound
of the expectation value of the square of the uncertainty in the amplitude
measurement does not become infinite when the total observational term
approaches zero. In order to sum over all possible momentum modes, we next
consider a noise term with a suitable Gaussian weight factor which decays over
time. We then obtain the lower bound on the final expectation value of the
square of the variance in the amplitude parameter. Because of the noise induced
by the graviton, there is a minimum value of the measurement time below which
it is impossible to detect any gravitational wave using a Bose-Einstein
condensate. Finally, we consider interaction between the phonon modes of the
Bose-Einstein condensate which results in the decoherence. We observe that the
decoherence effect becomes significant for gravitons with minimal squeezing.
