of the temporal delay $\tau$, by measuring the spectrum of the outgoing signal in the selected output port. The different spectra obtained for delays $\tau=0$ and $\tau=100$ as, for two different polarisation projections, are shown in Figures 2 (a) and 2 (b). To characterize different modes one can measure, for instance, the centroid of the spectrum. Fig. 2 (c) shows the centroid shift of the output signal for $\tau \ne 0$, which reads $$\Delta
f=-\frac{\tau\,\ln 2 }{\pi T_0^2}  \frac{\gamma \sin
\left(\omega_0 \tau - \Gamma\right)}{1+\gamma \cos \left(\omega_0
\tau-\Gamma\right)}, \label{centroid_shift}$$ The differential power between both signals (with $\tau=0$ and $\tau \ne 0$) reads $$\frac{P_{\mathrm{out}}(\tau)-P_{\mathrm{out}}(\tau=0)}{P_{\mathrm{in}}}=\frac{1}{2}\,
\left[ \cos \Gamma-\cos \left(\omega_0 \tau -\Gamma \right)
\right]$$ When there is no polarisation-dependent time delay ($\tau=0$), the centroid of the spectrum of the output signal is the same than the centroid of the input laser beam, i.e., there is no shift of the centroid ($\Delta f=0$). However, the presence of a small $\tau$ can produce a large and measurable shift of the centroid of the spectrum of the signal.

Results {#results .unnumbered}
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View of weak value amplification from quantum estimation theory {#view-of-weak-value-amplification-from-quantum-estimation-theory .unnumbered}
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Detecting the presence ($\tau \neq 0$) or absence ($\tau=0$) of a temporal delay 