Dynamics from noisy data with extreme timing uncertainty.
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 magnitude, 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 uncertainty. 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 analysis, 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 uncertainty. 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.
Department of Physics, University of Wisconsin Milwaukee, 3135 North Maryland Avenue, Milwaukee, Wisconsin 53211, USA.
R
R
Fung
Center for Free-Electron Laser Science, DESY, Notkestrasse 85, 22607 Hamburg, Germany.|The Hamburg Centre for Ultrafast Imaging, Luruper Chausee 149, 22761 Hamburg, Germany.|Department of Chemistry, University of Hamburg, Grindelallee 117, 20146 Hamburg, Germany.
A M
AM
Hanna
Center for Free-Electron Laser Science, DESY, Notkestrasse 85, 22607 Hamburg, Germany.|The Hamburg Centre for Ultrafast Imaging, Luruper Chausee 149, 22761 Hamburg, Germany.
O
O
Vendrell
Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA.
S
S
Ramakrishna
Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA.
T
T
Seideman
Center for Free-Electron Laser Science, DESY, Notkestrasse 85, 22607 Hamburg, Germany.|The Hamburg Centre for Ultrafast Imaging, Luruper Chausee 149, 22761 Hamburg, Germany.|Department of Chemistry, University of Hamburg, Grindelallee 117, 20146 Hamburg, Germany.|Department of Physics, University of Hamburg, Jungiusstrasse 9, 20355 Hamburg, Germany.
R
R
Santra
Department of Physics, University of Wisconsin Milwaukee, 3135 North Maryland Avenue, Milwaukee, Wisconsin 53211, USA.
A
A
Ourmazd
Department of Physics, University of Wisconsin Milwaukee, 3135 North Maryland Avenue, Milwaukee, Wisconsin 53211, USA.
A
A
Ourmazd