Estimation of displacement operations based on Fock states considering thermal noise in preparing probes

Displacement based quantum receivers can beat the standard quantum limit and have gained increasing attention in recent years. The displacement operation value can be estimated and calibrated by using Fock states as probes. In this Letter, we focus on the impacts of thermal noise when preparing the Fock probes on the estimation performance. We first derive the density matrix and the photon statistics of the noisy displaced number state with a uniformly distributed displacement phase. Based on the derived density matrix, we then analyze the performance of estimating displacement operations by using the maximum likelihood estimation (MLE) method. To decrease the computational complexity of the MLE, we further derive an approximated probability density of the photon numbers under small displacement. Besides, the quantum Cramér-Rao bound (QCRB) is also derived to analyze the performance limit of the displacement estimation. Numerical results found that, though thermal noise can greatly degrade the MLE accuracy of the displacement operations, the performance advantage of using Fock probes still persists over classical coherent probes. Besides, we demonstrate that the MLE method based on Fock probes can well approach the QCRB when the displacement strength N ≤ ~1.