Ultrasensitive biased weak measurement for longitudinal phase estimation

Standard weak measurement (SWM) has been proved to be a useful ingredient for measuring small longitudinal phase change (LPC) [X. Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C. F. Li, and Guang-Can Guo, Phys. Rev. Lett. 111, 033604 (2013), 10.1103/PhysRevLett.111.033604]. In these SWM proposals, the working regime is selected where the relative phase between orthogonal components is zero, and the LPC can be detected by measuring the shift of the meter state, i.e., the time and frequency domain, which are conjugated observables. Here, we show that an extra bias phase can significantly improve the sensitivity of measuring LPC. This bias phase can be introduced by a precoupling process. Together with a special postselection, a destructive interference can be observed in both the time and frequency domain. Using a broadband source, this conjugated destructive interference occurs in a regime less than 1 as, while the related spectral shift reaches hundreds of THz, resulting in sensitivity outperforming SWM by two orders of magnitude.