Imperfect knowledge of the times at which ‘snapshots’ of a system are recorded degrades our ability to recover dynamical information, and can scramble the sequence of events. In X-ray free-electron lasers, for example, the uncertainty—the so-called timing jitter—between the arrival of an optical trigger (‘pump’) pulse and a probing X-ray pulse can exceed the length of the X-ray pulse by up to two orders of magnitude1, marring the otherwise precise time-resolution capabilities of this class of instruments. The widespread notion that little dynamical information is available on timescales shorter than the timing uncertainty has led to various hardware schemes to reduce timing uncertainty2,3,4. These schemes are expensive, tend to be specific to one experimental approach and cannot be used when the record was created under ill-defined or uncontrolled conditions such as during geological events. Here we present a data-analytical approach, based on singular-value decomposition and nonlinear Laplacian spectral analysis5,6,7, that can recover the history and dynamics of a system from a dense collection of noisy snapshots spanning a sufficiently large multiple of the timing uncertainty. The power of the algorithm is demonstrated by extracting the underlying dynamics on the few-femtosecond timescale from noisy experimental X-ray free-electron laser data recorded with 300-femtosecond timing uncertainty1. Using a noisy dataset from a pump-probe experiment on the Coulomb explosion of nitrogen molecules, our analysis reveals vibrational wave-packets consisting of components with periods as short as 15 femtoseconds, as well as more rapid changes, which have yet to be fully explored. Our approach can potentially be applied whenever dynamical or historical information is tainted by timing uncertainty.
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We thank P. Bucksbaum, J. M. Glownia and A. Natan for experimental data and comments on an earlier version of the manuscript, and acknowledge discussions with A. Dashti, D. Giannakis, A. Hosseinizadeh, A. Rudenko, M. Schmidt and P. Schwander. The research conducted by A.O. and R.F. was supported by the US Department of Energy, Office of Science, Basic Energy Sciences under award DE-SC0002164 (algorithm design and development, and data analysis), and by the US National Science Foundation under awards STC 1231306 (numerical trial models) and 1551489 (underlying analytical models). The research conducted by T.S. and S.R. was supported by the US Department of Energy, Office of Science, Basic Energy Sciences under award DE-FG02-04ER15612. T.S. thanks the Hamburg Centre for Ultrafast Imaging for a Mildred Dresselhaus Visiting Professorship.
The authors declare no competing financial interests.
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Fung, R., Hanna, A., Vendrell, O. et al. Dynamics from noisy data with extreme timing uncertainty. Nature 532, 471–475 (2016). https://doi.org/10.1038/nature17627
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