Measuring Ultrasmall Time Delays of Light by Joint Weak Measurements

We propose to use weak measurements away from the weak-value amplification regime to carry out precision measurements of time delays of light. Our scheme is robust to several sources of noise that are shown to only limit the relative precision of the measurement. Thus, they do not set a limit on the smallest measurable phase shift, contrary to standard interferometry and weak-value-based measurement techniques. Our idea is not restricted to phase-shift measurements and could be used to measure other small effects using a similar protocol.

Figure 1

(a) Mach-Zehnder interferometer for laser light with a time-delay asymmetry $\tau$. The incoming beam is linearly polarized and then evenly split by a polarizing beam splitter (PBS) into the two arms with opposite circular polarizations, ${\sigma }_y=\pm 1$. The beams are recombined at a second PBS with linearly polarized outputs. The direction of the linear polarization is described by the azimuthal angle $\varphi$ on the Poincaré sphere. Two spectrometers labeled by $q=\pm 1$ measure the spectrum of the outgoing light. (b) Graphical representation of the estimation process. The values of $\tau$ and $\varphi$ are extracted from the spectra measured by the two detectors by a maximum-likelihood (ML) estimation technique, as described in the text.

Figure 2

Ultimate precision limit $\Delta {\tau }_{\mathrm{ult}}$ on the estimation of the physical time delay $\tau$ as a function of $\tau$, in the presence of fluctuations of the alignment $\varphi$; see Eq. (4). We assume $ϵ=0.02\mathrm{rad}$ for the plot. The dashed line shows the constant (zeroth-order) contribution of fluctuations to $\Delta {\tau }_{\mathrm{ult}}=Cϵ$ for weak-value amplification; here, we assume $C\approx 0.25\times {10}^{-18}\mathrm{s}$, as in Ref. 5. The solid line is the result of our scheme, $\Delta {\tau }_{\mathrm{ult}}={ϵ}^2\tau /2$. For $\tau <2C/ϵ$ (shaded area), our scheme outperforms the weak-value amplification.