Measurement of sub-pulse-width temporal delays via spectral interference induced by weak value amplification

We demonstrate experimentally a scheme to measure small temporal delays, much smaller than the pulse width, between optical pulses. Specifically, we observe an interference effect, based on the concepts of quantum weak measurements and weak value amplification, through which a sub-pulse-width temporal delay between two femtosecond pulses induces a measurable shift of the central frequency of the pulse. The amount of frequency shift, and the accompanying losses of the measurement, can be tailored by postselecting different states of polarization. Our scheme requires only spectrum measurements and linear optics elements, hence greatly facilitating its implementation. Thus it appears to be a promising technique for measuring small and rapidly varying temporal delays.

Figure 1

Schematic of the experimental setup. State preselection: The polarization of the input optical pulse is selected by using $\lambda /2$ and $\lambda /4$ wave plates (not shown). Weak coupling: A Michelson-Morley interferometer, composed of a polarizing beam splitter (PBS), two $\lambda /4$ wave plates (QWP), and two mirrors, divides the input pulse into two pulses, with equal power and with orthogonal polarizations, that travel through different paths of the interferometer. Each mirror is mounted in a translation stage that allows changing the temporal delay in each path. State postselection: The two pulses recombine in the PBS, and they are projected into a particular state of polarization with a electrically controlled liquid crystal variable retarder (LCVR) and a polarizer. The output beam is finally focused in a single-mode fiber (SM) and its spectrum is measured with an optical spectrum analyzer (OSA).

Figure 2

Measurement of the central frequency shift induced by weak value amplification. Measured frequency shift (a) and insertion loss (b) as a function of the postselection angle $\Gamma$. Dots (with error bars) are experimental results, and the dotted lines are best theoretical fits using the measured input spectrum in Eq. (2). The best fits are obtained for $T=53\mathrm{fs}$ in panels (a) and (b) and $T=22\mathrm{fs}$ in panels (i) and (j). For $T=53\mathrm{fs}$, panels (c), (e), and (g) (measured) and (d), (f), and (h) (theory) show the spectral density for some selected cases, as indicated by the corresponding labels in panels (a) and (b). For $T=22\mathrm{fs}$, panels (k), (m), and (o) (measured) and (l), (n), and (p) (theory) show the spectral density for some selected cases, as indicated by the corresponding labels in panels (i) and (j). To help the eye, the central frequency of the input pulse (${\nu }_0=193.44\mathrm{THz}$) is represented by a dashed line in all plots. The experiment is performed at a temperature of $34.1{}^{\circ }\text{C}$. Error bars in all plots assume that temperature variations during the experiment are in the range of $\pm 1{}^{\circ }\text{C}$, which translates in random changes of the angle of postselection $\Gamma$.

Figure 3

Feasibility of the measurement of attosecond temporal delays with femtosecond pulses. Polarization-dependent frequency shift induced by a $T=10$ as time delay of pulses of duration $\tau =10$ fs. (a) Low-loss and (b) high-loss regime. The solid line (blue [gray]) corresponds to the frequency shift and the dashed (green [gray]) line to the insertion losses. Notice the different scales in the $x$ and $y$ axes in panels (a) and (b).