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Physical Review A

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

John C. Howell

Published Jul 29, 2022• 1 citation
Publisher

Abstract

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 ${10}^{\ensuremath{-}34}$ 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.

DOI
10.1103/physreva.106.012224

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Physica Scripta•Oct 16, 2026•0 citations
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\ensuremath−34{10}^{\ensuremath{-}34}10\ensuremath−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.
In the standard weak measurement, the imaginary part of the weak value is usually constructed with a specific pre- or postselection state to cause a large shift of the frequency-domain pointer. In this paper we demonstrate that the purely real weak value can also shift the frequency-domain pointer when using a broad-spectrum light source. The reason for the deflection of the frequency-domain pointer in this scheme is that the postselection probability of different frequency is different, which causes the rearrangement of the amplitudes of the spectrum. The weak measurement with real weak value can be used for the estimation of time delay, and it is found that it can reach the same high resolution as the standard weak measurement in some sensitive regions.
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.
In quantum weak measurement, a weak value is a quantity related to a pointer shift of a measuring device which is usually determined by pre- and postselection states. For conventional weak measurement, a single weak-value model has been well addressed in precision metrology of time delay, but it only holds for a narrowband light source. To further improve the measurement accuracy, a broad-spectrum light source is needed. We find that the single weak-value model no longer holds, due to the occurrence of modified preselection states. Here, a multiple weak-value model is established by introducing the modified preselection states in the estimation of time delay. For broad-spectrum light source with 50-nm spectral width, the multiple weak-value scheme enables us to achieve a high resolution of $1.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\phantom{\rule{0.16em}{0ex}}\mathrm{as}$. Our proposed multiple weak-value amplification may have important applications in various fields involving precise time-delay detection.
Background: Weak measurement involves weak coupling between the system and the measuring device (pointer) enables large amplification and high precision measurement of small physical parameters. The outcome of this special measurement procedure involving nearly mutually orthogonal pre- and post-selection of states in such weakly interacting systems leads to weak value that can become exceedingly large and lie outside the eigenvalue spectrum of the measured observable. This unprecedented ability of weak value amplification of small physical parameters has been successfully exploited for various metrological applications in the optical domain and beyond. Even though it is a quantum mechanical concept, it can be understood using the classical electromagnetic theory of light and thus can be realized in classical optics. Objective: Here, we briefly review the basic concepts of weak measurement and weak value amplification, provide illustrative examples of its implementation in various optical domains. The applications involve measuring ultra-sensitive beam deflections, high precision measurements of angular rotation, phase shift, temporal shift, frequency shift and so forth, and expand this extraordinary concept in the domain of nano-optics and plasmonics. Methods: In order to perform weak value amplification, we have used Gaussian beam and spectral response as the pointer subsequently. The polarization state associated with the pointer is used as pre and post-selection device. Results: We reveal the weak value amplification of sub-wavelength optical effects namely the Goos-Hänchen shift and the spin hall shift. Further, we demonstrate weak measurements using spectral line shape of resonance as a natural pointer, enabling weak value amplification beyond the conventional limit, demonstrating natural weak value amplification in plasmonic Fano resonances and so forth. The discussed concepts could have useful implications in various nano-optical systems to amplify tiny signals or effects. Conclusion: The emerging prospects of weak value amplification towards the development of novel optical weak measurement devices for metrological applications are extensively discussed.
Abstract When a polarized optical pulse propagates through a Faraday medium, the two circular polarization eigenmodes accumulate differential phase shifts and differential group delay. These two eigenmodes thus become distinguishable by the arrival time of photons, which can serve as a pointer state in the weak-measurement (WM) framework. Here, we describe the Faraday effect as a von Neumann-type interaction, coupling the polarization degree of freedom of light to the temporal envelope of the pulse. Under suitable pre- and post-selection of nearly orthogonal linear polarizations, the two delayed circular components interfere spectrally. This produces an amplified spectral centroid shift, corresponding to the imaginary weak value of the circular polarization component, which is highly sensitive to the small phase shifts induced by the Faraday effect. We propose detecting this spectral shift using heterodyne detection by measuring the centroid of the radio-frequency beat spectrum, a method suitable for atomic magnetometry. While this single-pulse approach may be viable where steep dispersion near resonance yields considerable group delay, it would be ineffective in most far-off-resonant magnetometry where group delay is insignificant. We propose to overcome this challenge by employing two oppositely circularly polarized pulses with a controllable phase difference and a synthetic temporal delay. These engineered input pulses provide tunable sensitivity by allowing weak-value amplification for detecting small magnetic field variations, even in the presence of an ambient magnetic field. Crucially, because the post-selection process prevents detector saturation, it allows for the use of significantly higher optical power, up to the atomic saturation level. For far-off-resonant Faraday magnetometry, we estimate that this detection scheme could potentially allow 10–100 femtotesla-level ( fT Hz − 1 ) photon-shot-noise-limited sensitivity for thermal alkali-vapors depending on detuning, with microtesla-level tunable dynamic range. Hence, this WM-based heterodyne spectral centroid readout scheme can mitigate technical noise, paving the way for high-sensitivity alkali-vapor magnetometry in unshielded environments.