  Standard weak measurement (SWM) has been proved to be a useful ingredient for
measuring small longitudinal phase shifts. [Phys. Rev. Lett. 111, 033604
(2013)]. In this letter, we show that with specfic pre-coupling and
postselection, destructive interference can be observed for the two conjugated
variables, i.e. time and frequency, of the meter state. Using a broad band
source, this conjugated destructive interference (CDI) can be observed in a
regime approximately 1 attosecond, while the related spectral shift reaches
hundreds of THz. This extreme sensitivity can be used to detect tiny
longitudinal phase perturbation. Combined with a frequency-domain analysis,
conjugated destructive interference weak measurement (CDIWM) is proved to
outperform SWM by two orders of magnitude.
