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Physical Review A

Precision estimation of time delay based on weak measurement with real weak value

Weining Liu, Yisen Wang, Hailu Luo

Published Nov 13, 2023• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.108.052606

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Physical Review A•Aug 4, 2025•0 citations
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.
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.
We propose a scheme called the weak measurement with two pointers to measure time delays in a large dynamic range with high precision. One pointer with the light frequency and the other pointer with the postselected probability play a complementary role in the dynamic range of the time-delay estimation. By inserting the main systematic errors, we study the antierror capability and the amplification behavior of two types of pointers, where the pointer with the postselected probability shows stronger robustness against systematic errors and larger working range in general. The proof-of-principle experiments together with the Fisher information metric analysis show that high precision can be effectively retained in a large dynamic range of the time-delay estimation. Further implementation strategy of this scheme is also provided, which may promote the precise estimation of other parameters in a large dynamic range.
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.
Weak measurement enables the amplification of weak physical effects via post-selection and has become an important tool in precision optical metrology; however, conventional schemes based on mean-pointer shifts suffer from response saturation, limited linear range, and stringent stability requirements. Here, we propose and experimentally demonstrate a weak-measurement scheme based on spectral-interference trough shifts, where the zero-intensity points of the post-selected spectrum act as the measurement pointer, establishing an analytical mapping between the trough displacement and the target phase or time delay. Theoretical analysis shows that, under detector resolution limits, the measurement resolution depends solely on the frequency of extinction point and is independent of weak-value singular amplification or bias-phase modulation, thereby maintaining high sensitivity while avoiding pointer saturation. Experiments demonstrate that the trough-shift scheme achieves significantly better agreement between measured and theoretical sensitivities than biased weak measurement and provides a stable linear response without additional bias-compensation structures, reaching a minimum resolvable phase variation at the 10−7 level. Moreover, the approach intrinsically supports multi-period traceable measurements and exhibits strong robustness against intensity fluctuations and spectral distortions, offering a promising route toward high-sensitivity, large-dynamic-range, and stable weak measurement-based optical sensing.
Weak measurements with imaginary weak values are reexamined in light of recent experimental results. The shift of the meter, due to the imaginary part of the weak value, is derived via the probability of postselection, which allows considering the meter as a distribution of a classical variable. The derivation results in a simple relation between the change in the distribution and its variance. By applying this relation to several experimental results, in which the meter involved the time and frequency domains, it is shown to be especially suitable for scenarios of that kind. The practical and conceptual implications of a measurement method, which is based on this relation, are discussed.
Optical time delays play a pivotal role in modern physics and practical applications. We propose and experimentally validate a weak measurement system with a double-edge filter, which enables the real-time, high-precision estimation of varying time delays, and the effective suppression of technical noise. In the experiment, the postselected probe light is processed by the double-edge filter, which is formed by a pair of fiber Bragg gratings positioned symmetrically about the center frequency of the probe light, and subsequently detected by photodetectors. The experimental results demonstrate a time-delay precision of 1.66 attoseconds (as) and real-time estimation capability. With a postselection angle of ${10}^{\ensuremath{-}3}\phantom{\rule{0.16em}{0ex}}\mathrm{rad}$ and the system operating within the most sensitive linear dynamic range, the precision can be further improved to ${10}^{\ensuremath{-}4}$ as. The approach benefits from its high sensitivity and rapid detection, which shows great potential applications in real-time and high-precision time-delay measurements.