Overcoming Residual Timing Jitter in Pump-Probe Interferometry via Weak Value Amplification and Deep Learning

TL;DR

We address residual timing jitter in pump-probe interferometry aiming at attosecond-resolution time delays and propose a hybrid approach that couples weak value amplification with deep learning to recover tiny delays from jittered 2D interferograms. Real weak values $A_w$ amplify the pump-induced delay $\tau$, while CNN-based estimators extract the delay and are benchmarked against FFT analysis, under realistic jitter $\delta t_{jitter}$. Results show that WVA improves SNR and both CNN-Regressor and CNN-Classifier outperform FFT, with the former excelling at small $A_w$ and the latter at large $A_w$, including the challenging $\pi$ phase-shift regime; performance approaches the classical Fisher information bound as $\tau$ increases. The approach advances attosecond metrology by enabling near-CFI-limited delay estimation and can be extended to other ultrafast metrology tasks requiring robust reconstruction of subtle, noisy signals.

Abstract

We introduce a hybrid methodology that synergistically combines weak value amplification (WVA) and deep learning to suppress the limiting effects of residual timing jitter in pump-probe interferometry, achieved through simulations of pump-induced time delays at a few-attosecond resolution. The WVA protocol, employing real weak values, amplifies the minute delay induced by sample perturbation, thereby translating it into a measurable shift of interference fringes. However, this amplification introduces significant fringe distortion. To address this, we deploy deep learning architectures as high-precision parameter estimators: a convolutional neural network regressor (CNN-Regressor) for direct delay estimation and a classifier (CNN-Classifier) for discrete delay categorization. These are systematically benchmarked against traditional Fourier-transform-based analysis. Two key conclusions are drawn: (i) The WVA technique consistently enhances measurement precision across all estimators by effectively increasing the signal-to-noise ratio (SNR). (ii) Both deep learning models surpass the traditional FFT approach; the CNN-Regressor achieves a higher SNR at small weak values, while the CNN-Classifier enables accurate estimation even under a challenging "$π$ phase shift" condition where conventional analysis fails. This synergistic combination of WVA and deep learning establishes a powerful framework for attosecond metrology, paving the way for enhanced precision in ultrafast spectroscopy.

Figures (5)

• Figure 1: Schematic diagrams of pump-probe interferometry configurations. (a) Traditional pump-probe interferometry setup. The diagrams are adapted from Ref. Tanghe2022. (b) Weak value amplification (WVA)-enhanced pump-probe interferometry implementation. The reference sample is subjected to the same pump pulse with identical timing jitter $\delta t_{\text{jitter}}$ as the main sample path, ensuring selective amplification of the sample-induced time delay $\tau$ while rejecting timing jitter (see main text for details). Key components: Ultrafast Laser (generating Fourier-transform-limited Gaussian pulses); P1, P2 (polarizers); M1-M5 (mirrors); L1-L3 (lenses); PBS1, PBS2 (polarization beam splitters); Grating (diffraction grating); CCD ( CCD camera).
• Figure 2: Simulated interferograms and deep learning architecture. (a) Spatial-spectral interferograms for pump-induced delays $\tau$ (0 as $< \tau <$ 10 as) at varying weak values $A_w$. Horizontal pixels correspond to spectral information (photon energy), while vertical pixels encode spatial information along the $y$-direction. (b) Deep convolutional neural network architecture employed for parameter estimation in WVA-enhanced pump-probe interferometry.
• Figure 3: Traditional FFT analysis of fringe shifts. Fringe shift measurements derived via FFT-based subpixel registration for pump-induced delays $\tau$ at varying weak values $A_w$. Note the emergence of $\pi$ phase shift discontinuities in panels (c) and (d), which violate the linear phase assumption of conventional interferometric analysis and introduce significant estimation errors.
• Figure 4: Deep learning performance for time delay estimation. (a)-(d) Comparison between predicted and actual time delays $\tau$ across varying $A_w$ values. (e)-(h) Signal-to-noise ratio performance relative to the classical Fisher information (CFI) bound and the traditional FFT approach. Each data point represents a single measurement instance. The SNR values for the traditional FFT approach were calculated from the data in Fig. \ref{['Fig:FFT_reslut']} as the mean divided by the standard deviation of the fringe shift measurements.
• Figure 5: Estimator for the WVA-based pump-probe interferometry: deep convolutional neural networks. (a) Root mean square error (RMSE) in various $A_w$ as a function of the iteration number within the CNN-Regressor network. (b) The accuracy of the training on various $A_w$ as a function of the number of iterations within the CNN classifier network.