Standard quantum limit of finite-size optical lattice clock in estimating gravitational potential

We evaluated the accuracy limit for estimating gravitational potential using optical lattice clocks by utilizing the quantum Cram\'{e}r--Rao bound. We then compared the results for single-layer and multilayer optical lattice clocks. The results indicate that the lower bound of variance of the estimator of gravitational potential using finite-size optical lattice clocks diverges and recovers repeatedly as a function of time. Namely, the accuracy of the gravitational potential estimation is not a monotonic function of time owing to the effect of gravitational dephasing in finite-size optical lattice clock. Further, this effect creates an estimation accuracy limit when attempting to avoid the divergence of the lower bound. When the number of layers in the optical lattice clock is sufficiently large, the limit is independent of the optical lattice clock details. The time required to reach this limit is calculated to be approximately 33 hours for a three-dimensional optical lattice clock consisting of one million cadmium atoms due to Earth's gravity, and approximately the same for other atoms.