Weakness of weak values: Incompatibility of anomalous pulse-spectrum amplification and optical frequency combs

We probe the use of optical frequency combs to place lower bounds on anomalous amplification of a weak-value-measured pulse delay, potentially reaching a theoretical temporal resolution of better than ${10}^{-34}$ s. Owing to the interferometric behavior of weak values, we show that anomalous weak value amplification of a time delay is not equivalent to a temporal linear phase ramp. We show that the anomalous weak value is a rearrangement of amplitudes that generates an apparent shift that can be measured in direct detection, but does not change the actual frequency offset of a spectral distribution measurable in coherent detection. This implies that high-precision heterodyne beatnote frequency measurements of an optical frequency comb cannot be used in combination with the weak value amplification, at least in its currently conceived form. This unfortunate feature sheds light on the nature of anomalous weak values in shifting the pointer in a conjugate basis.

Figure 1

Experimental schematic. An optical frequency comb spectrum is placed in an anomalous weak value configuration. A preselected state is chosen using a combination of a polarizer (Pol) and quarter wave plate ($\lambda /4$). A birefringent crystal (BC) imparts a relative phase time delay for each polarization. For simplicity, it is assumed that no transverse walk-off occurs in the crystal. The postselection is made with a second quarter wave plate and polarizer.

Figure 2

Graphical depiction of input and output spectrum of the anomalous weak values interferometer. The comb teeth have not shifted, but the amplitudes have changed. It should be noted that the output spectrum will be considerably weaker than the input spectrum.