Optical Estimation of Displacement Operation Based on Photon Number States Under Thermal Noise

Displacement based quantum receivers have gained increasing attention because they can beat the standard quantum limit. The displacement value in quantum receivers can be calibrated by using number states as probes. Current estimating methods assumed that ideal number states can be prepared perfectly, which is impossible in practical implementations. In this work, we focus on the impacts of thermal noise when preparing the number states on the estimation performance of displacement operations. We first derive the probability density of photon numbers in a thermal noise contaminated number state after a displacement operation, which we call the noisy displaced number state (NDNS). Based on the derived probability density, we then analyze the performance of estimating displacement operations with NDNS probes by using the maximum likelihood estimation (MLE) method. Numerical results found that the thermal noise in preparing the number states has great influence on the MLE accuracy of the displacement operations.