The positive role of gain noise in non-Hermitian reciprocal quantum sensing with two coherent drives

Unconventional non-Hermitian properties have been harnessed to enhance quantum sensing. Non-reciprocity has been identified as a powerful resource for non-Hermitian sensing. However, it was demonstrated that with two coherent drives, reciprocal non-Hermitian sensors can attain the same limit as those of non-reciprocal sensors. In this paper we demonstrate the positive role played by the gain noise in making a reciprocal sensor to have an arbitrarily large sensitivity. To do this, we take the best possible measurement rate per photon as the figure of merit. It is found that for the single-drive case, the best possible measurement rate per photon of reciprocal and non-reciprocal sensors do not change in the ideal case where the gain and loss are ignored; For the two-drive case, the best performance of non-reciprocal sensors can still be made arbitrarily large no matter whether gain and loss are involved. The gain noise is however crucial for reciprocal sensors to have an arbitrarily large sensitivity. Usually the noise is regarded as a negative effect that can greatly degrade the performance of sensing, this work helps to better understand the positive role of gain in reciprocal sensing.