  In this paper we have considered a quantized and linearly polarized
gravitational wave interacting with a gravitational wave detector
(interferometer detector) in the generalized uncertainty principle (GUP)
framework. Following the analysis in Phys. Rev. Lett. 127 (2021) 081602
(https://link.aps.org/doi/10.1103/PhysRevLett.127.081602), we consider a
quantized gravitational wave interacting with a gravitational wave detector
(LIGO/VIRGO etc.) using a path integral approach. Although the incoming
gravitational wave was quantized, no Planck-scale quantization effects were
considered for the detector in earlier literatures. In our work, we consider a
modified Heisenberg uncertainty relation with a quadratic order correction in
the momentum variable between the two phase space coordinates of the detector.
Using a path integral approach, we have obtained a stochastic equation
involving the separation between two point-like objects. It is observed that
random fluctuations (noises) and the correction terms due to the generalized
uncertainty relation plays a crucial role in dictating such trajectories.
Finally, we observe that the solution to the stochastic equation leads to time
dependent standard deviation due to the GUP insertion, and for a primordial
gravitational wave (where the initial state is a squeezed state) both the noise
effect and the GUP effects exponentially enhance which may be possible to
detect in future generation of gravitational wave detectors. We have also given
a plot of the dimensionless standard deviation with time depicting that the GUP
effect will carry a distinct signature which may be detectable in the future
space based gravitational wave observatories.
