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Separating single- from multi-particle dynamics in nonlinear spectroscopy

An Author Correction to this article was published on 01 August 2023

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Quantum states depend on the coordinates of all their constituent particles, with essential multi-particle correlations. Time-resolved laser spectroscopy1 is widely used to probe the energies and dynamics of excited particles and quasiparticles such as electrons and holes2,3, excitons4,5,6, plasmons7, polaritons8 or phonons9. However, nonlinear signals from single- and multiple-particle excitations are all present simultaneously and cannot be disentangled without a priori knowledge of the system4,10. Here, we show that transient absorption—the most commonly used nonlinear spectroscopy—with N prescribed excitation intensities allows separation of the dynamics into N increasingly nonlinear contributions; in systems well-described by discrete excitations, these N contributions systematically report on zero to N excitations. We obtain clean single-particle dynamics even at high excitation intensities and can systematically increase the number of interacting particles, infer their interaction energies and reconstruct their dynamics, which are not measurable via conventional means. We extract single- and multiple-exciton dynamics in squaraine polymers11,12 and, contrary to common assumption6,13, we find that the excitons, on average, meet several times before annihilating. This surprising ability of excitons to survive encounters is important for efficient organic photovoltaics14,15. As we demonstrate on five diverse systems, our procedure is general, independent of the measured system or type of observed (quasi)particle and straightforward to implement. We envision future applicability in the probing of (quasi)particle interactions in such diverse areas as plasmonics7, Auger recombination2 and exciton correlations in quantum dots5,16,17, singlet fission18, exciton interactions in two-dimensional materials19 and in molecules20,21, carrier multiplication22, multiphonon scattering9 or polariton–polariton interaction8.

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Fig. 1: Impact of excitation intensity on TA of squaraine polymers.
Fig. 2: Single-exciton dynamics in squaraine polymers.
Fig. 3: High nonlinear orders and multi-exciton dynamics.
Fig. 4: Highly nonlinear TA applied to diverse samples.

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Data availability

The data supporting the findings of this study are given in graphic form in the paper (including the Supplementary Information).  Source data are provided with this paper. Additional raw data are available from the corresponding authors on reasonable request.

Code availability

All codes needed to evaluate the conclusions in the paper are described in the paper and the source code for the numerical calculations is available openly online55.

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We are indebted to H. Lokstein for providing the LHCII sample and to D. Hiller and F. Trojánek for letting us measure the silicon nanocrystal sample. We thank S. Büttner who measured cresyl violet on the new 100 kHz setup. We thank G. R. Fleming for reading the manuscript and useful suggestions. The work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) grant no. 423942615 to T.B. and by the SolTech Initiative of the Bavarian State Ministry of Education, Culture, Science and the Arts (C.L. and T.B.). J.K. and P.R. were supported by the Natural Science and Engineering Research Council of Canada (NSERC). J.L. acknowledges support by the Cusanuswerk and P.M. was supported by the Alexander von Humboldt Foundation.

Author information

Authors and Affiliations



T.B., P.M., J.L., P.A.R. and J.J.K. conceived the project. A.T. and C.L. synthesized the squaraine polymers used for measurement. P.M. and J.L. designed the experiments and together with P.A.R. analysed the results. P.M., P.A.R. and J.J.K. formulated the theoretical description of the data. All authors discussed the results. T.B. and J.J.K. supervised the project. P.M. and T.B. wrote the manuscript with input from all co-authors.

Corresponding authors

Correspondence to Pavel Malý, Jacob J. Krich or Tobias Brixner.

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Extended data figures and tables

Extended Data Fig. 1 Third-order signals of diverse samples.

a, Light-harvesting complex II. b, Cresyl violet dye. c, CdSe/Zns core-shell quantum dots and d, silicon nanocrystals. For all samples their structure is shown at the top part of the panels. At the bottom part the transient signal maps are shown. The transient maps contain regions indicated with solid, dashed, and dotted borders whose spectrally integrated signals are shown as solid, dashed, and dotted curves on top. Blue: PP(3) signals, black: low-power reference measurement.

Source data

Extended Data Fig. 2 Scheme of the experimental setup used for the squaraine polymer measurement.

The probe beam was attenuated by a filter wheel; the intensity of the pump beam was adjusted using the Dazzler pulse shaper. The half-wave plate \(\frac{\lambda }{2}\) was used to rotate the pump polarization to magic angle between the pump and probe polarization to measure the isotropic signal.

Supplementary information

Supplementary Information

Supplementary Sections 1–8, including Figs. 1–23 and Tables 1–5.

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Malý, P., Lüttig, J., Rose, P.A. et al. Separating single- from multi-particle dynamics in nonlinear spectroscopy. Nature 616, 280–287 (2023).

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