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, excitons4-6, 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.

Center for Nanosystems Chemistry Universität Würzburg Würzburg Germany

Department of Physics University of Ottawa Ottawa Ontario Canada

Faculty of Mathematics and Physics Charles University Prague Czech Republic

Institut für Organische Chemie Universität Würzburg Würzburg Germany

Institut für Physikalische und Theoretische Chemie Universität Würzburg Würzburg Germany

School of Electrical Engineering and Computer Science University of Ottawa Ottawa Ontario Canada

Erratum v

Nature. 2023 Aug;620(7976):E27. doi: 10.1038/s41586-023-06476-9. PubMed

Zobrazit více v PubMed

Mukamel, S. Principles of Nonlinear Optical Spectroscopy (Oxford Univ. Press, 1995).

Klimov, V. I., Mikhailovsky, A. A., McBranch, D. W., Leatherdale, C. A. & Bawendi, M. G. Quantization of multiparticle Auger rates in semiconductor quantum dots. Science 287, 1011–1013 (2000). PubMed DOI

Almand-Hunter, A. E. et al. Quantum droplets of electrons and holes. Nature 506, 471–475 (2014). PubMed DOI

Valkunas, L., Trinkunas, G., Liuolia, V. & van Grondelle, R. Nonlinear annihilation of excitations in photosynthetic systems. Biophys. J. 69, 1117–1129 (1995). PubMed DOI PMC

Stone, K. W. et al. Two-quantum 2D FT electronic spectroscopy of biexcitons in GaAs quantum wells. Science 324, 1169–1173 (2009). PubMed DOI

Kriete, B. et al. Interplay between structural hierarchy and exciton diffusion in artificial light harvesting. Nat. Commun. 10, 4615 (2019). PubMed DOI PMC

You, C., Nellikka, A. C., De Leon, I. & Magaña-Loaiza, O. S. Multiparticle quantum plasmonics. Nanophotonics 9, 1243–1269 (2020). DOI

Sun, Y. et al. Direct measurement of polariton–polariton interaction strength. Nat. Phys. 13, 870–875 (2017). DOI

Giura, P. et al. Multiphonon anharmonicity of MgO. Phys. Rev. B 99, 220304 (2019). DOI

Joo, T., Jia, Y., Yu, J., Lang, M. J. & Fleming, G. R. Third‐order nonlinear time domain probes of solvation dynamics. J. Chem. Phys. 104, 6089–6108 (1996). DOI

Malý, P. et al. From wavelike to sub-diffusive motion: exciton dynamics and interaction in squaraine copolymers of varying length. Chem. Sci. 11, 456–466 (2020). DOI

Völker, S. F. et al. Singlet–singlet exciton annihilation in an exciton-coupled squaraine-squaraine copolymer: a model toward hetero-J-aggregates. J. Phys. Chem. C 118, 17467–17482 (2014). DOI

Rehhagen, C. et al. Exciton migration in multistranded perylene bisimide J-aggregates. J. Phys. Chem. Lett. 11, 6612–6617 (2020). PubMed DOI

Tzabari, L., Zayats, V. & Tessler, N. Exciton annihilation as bimolecular loss in organic solar cells. J. Appl. Phys. 114, 154514 (2013). DOI

Steiner, F., Vogelsang, J. & Lupton, J. M. Singlet-triplet annihilation limits exciton yield in poly(3-hexylthiophene). Phys. Rev. Lett. 112, 137402 (2014). PubMed DOI

Zhu, H., Yang, Y. & Lian, T. Multiexciton annihilation and dissociation in quantum confined semiconductor nanocrystals. Acc. Chem. Res. 46, 1270–1279 (2013). PubMed DOI

Palato, S. et al. Investigating the electronic structure of confined multiexcitons with nonlinear spectroscopies. J. Chem. Phys. 152, 104710 (2020). PubMed DOI

Smith, M. B. & Michl, J. Singlet fission. Chem. Rev. 110, 6891–6936 (2010). PubMed DOI

Purz, T. L. et al. Coherent exciton–exciton interactions and exciton dynamics in a MoSe 2/WSe 2 heterostructure. Phys. Rev. B 104, L241302 (2021). DOI

Dostál, J. et al. Direct observation of exciton–exciton interactions. Nat. Commun. 9, 2466 (2018). PubMed DOI PMC

Heshmatpour, C. et al. Annihilation dynamics of molecular excitons measured at a single perturbative excitation energy. J. Phys. Chem. Lett. 11, 7776–7781 (2020). PubMed DOI

Ueda, A., Matsuda, K., Tayagaki, T. & Kanemitsu, Y. Carrier multiplication in carbon nanotubes studied by femtosecond pump-probe spectroscopy. Appl. Phys. Lett. 92, 233105 (2008). DOI

Bennett, D. I. G., Fleming, G. R. & Amarnath, K. Energy-dependent quenching adjusts the excitation diffusion length to regulate photosynthetic light harvesting. Proc. Natl Acad. Sci. USA 115, E9523–E9531 (2018). PubMed DOI PMC

Polman, A., Knight, M., Garnett, E. C., Ehrler, B. & Sinke, W. C. Photovoltaic materials: present efficiencies and future challenges. Science 352, aad4424 (2016). PubMed DOI

Müller, M. G. et al. Singlet energy dissipation in the photosystem II light-harvesting complex does not involve energy transfer to carotenoids. ChemPhysChem 11, 1289–1296 (2010). PubMed DOI

Auston, D. H., Shank, C. V. & LeFur, P. Picosecond optical measurements of band-to-band Auger recombination of high-density plasmas in germanium. Phys. Rev. Lett. 35, 1022–1025 (1975). DOI

Smith, G. O., Mayer, E. J., Kuhl, J. & Ploog, K. Pump-probe investigations of biexcitons in GaAs quantum wells. Solid State Commun. 92, 325–329 (1994). DOI

Smith, R. P. et al. Extraction of many-body configurations from nonlinear absorption in semiconductor quantum wells. Phys. Rev. Lett. 104, 247401 (2010). PubMed DOI

Sun, D. et al. Observation of rapid exciton–exciton annihilation in monolayer molybdenum disulfide. Nano Lett. 14, 5625–5629 (2014). PubMed DOI

Taguchi, S., Saruyama, M., Teranishi, T. & Kanemitsu, Y. Quantized Auger recombination of biexcitons in CdSe nanorods studied by time-resolved photoluminescence and transient-absorption spectroscopy. Phys. Rev. B 83, 155324 (2011). DOI

Chlouba, T. et al. Pathways of carrier recombination in Si/SiO

Pedersen, S., Baumert, T. & Zewail, A. H. Femtosecond real-time probing of reactions. 13. Multiphoton dynamics of mercury iodide (IHgI). J. Phys. Chem. 97, 12460–12465 (1993). DOI

Yokoyama, K., Silva, C., Son, D. H., Walhout, P. K. & Barbara, P. F. Detailed investigation of the femtosecond pump–probe spectroscopy of the hydrated electron. J. Phys. Chem. A 102, 6957–6966 (1998). DOI

Bittner, T., Irrgang, K.-D., Renger, G. & Wasielewski, M. R. Ultrafast excitation energy transfer and exciton–exciton annihilation processes in isolated light harvesting complexes of photosystem II (LHC II) from spinach. J. Phys. Chem. 98, 11821–11826 (1994). DOI

Brüggemann, B. & May, V. Exciton exciton annihilation dynamics in chromophore complexes. II. Intensity dependent transient absorption of the LH2 antenna system. J. Chem. Phys. 120, 2325–2336 (2004). PubMed DOI

Birkmeier, K., Hertel, T. & Hartschuh, A. Probing the ultrafast dynamics of excitons in single semiconducting carbon nanotubes. Nat. Commun. 13, 6290 (2022). PubMed DOI PMC

Kira, M., Koch, S. W., Smith, R. P., Hunter, A. E. & Cundiff, S. T. Quantum spectroscopy with Schrödinger-cat states. Nat. Phys. 7, 799–804 (2011). DOI

Leo, K. et al. Effects of coherent polarization interactions on time-resolved degenerate four-wave mixing. Phys. Rev. Lett. 65, 1340–1343 (1990). PubMed DOI

Tan, H.-S. Theory and phase-cycling scheme selection principles of collinear phase coherent multi-dimensional optical spectroscopy. J. Chem. Phys. 129, 124501 (2008). PubMed DOI

Brüggemann, B. & Pullerits, T. Nonperturbative modeling of fifth-order coherent multidimensional spectroscopy in light harvesting antennas. New J. Phys. 13, 025024 (2011). DOI

van Amerongen, H. & van Grondelle, R. Understanding the energy transfer function of LHCII, the major light-harvesting complex of green plants. J. Phys. Chem. B 105, 604–617 (2001). DOI

Barzda, V. et al. Singlet–singlet annihilation kinetics in aggregates and trimers of LHCII. Biophys. J. 80, 2409–2421 (2001). PubMed DOI PMC

Kostjukov, V. V. Photoexcitation of cresyl violet dye in aqueous solution: TD-DFT study. Theor. Chem. Acc. 140, 155 (2021). DOI

Diels, J.-C. & Rudolph, W. Ultrashort Laser Pulse Phenomena: Fundamentals, Techniques, and Applications on a Femtosecond Time Scale (Academic Press, 1996).

Yu, S., Titze, M., Zhu, Y., Liu, X. & Li, H. Observation of scalable and deterministic multi-atom Dicke states in an atomic vapor. Opt. Lett. 44, 2795–2798 (2019). DOI

Bangert, U., Bruder, L. & Stienkemeier, F. Pulse overlap ambiguities in multiple quantum coherence spectroscopy. Opt. Lett. 48, 538–541 (2023). PubMed DOI

Seiler, H., Palato, S. & Kambhampati, P. Investigating exciton structure and dynamics in colloidal CdSe quantum dots with two-dimensional electronic spectroscopy. J. Chem. Phys. 149, 074702 (2018). PubMed DOI

Sewall, S. L., Cooney, R. R., Anderson, K. E. H., Dias, E. A. & Kambhampati, P. State-to-state exciton dynamics in semiconductor quantum dots. Phys. Rev. B 74, 235328 (2006). DOI

Chlouba, T. et al. Interplay of bimolecular and Auger recombination in photoexcited carrier dynamics in silicon nanocrystal/silicon dioxide superlattices. Sci. Rep. 8, 1703 (2018). PubMed DOI PMC

Biggs, J. D., Voll, J. A. & Mukamel, S. Coherent nonlinear optical studies of elementary processes in biological complexes: diagrammatic techniques based on the wave function versus the density matrix. Philos. Trans. R. Soc. A 370, 3709 (2012). DOI

Abramavičius, D. Revealing a full quantum ladder by nonlinear spectroscopy. Lith. J. Phys. 60, 154–166 (2020).

Liu, Z. et al. Crystal structure of spinach major light-harvesting complex at 2.72 Å resolution. Nature 428, 287–292 (2004). PubMed DOI

van Grondelle, R. & Novoderezhkin, V. I. Energy transfer in photosynthesis: experimental insights and quantitative models. Phys. Chem. Chem. Phys. 8, 793–807 (2006). PubMed DOI

Trebino, R. Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Springer, 2002).

Krich, J. J., Rose, P. A. & Malý, P. Software for “Separating single- from multi-particle dynamics in nonlinear spectroscopy”. Zenodo https://doi.org/10.5281/zenodo.7675564 (2023).