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Precision test of statistical dynamics with state-to-state ultracold chemistry


Chemical reactions represent a class of quantum problems that challenge both the current theoretical understanding and computational capabilities1. Reactions that occur at ultralow temperatures provide an ideal testing ground for quantum chemistry and scattering theories, because they can be experimentally studied with unprecedented control2, yet display dynamics that are highly complex3. Here we report the full product state distribution for the reaction 2KRb → K2 + Rb2. Ultracold preparation of the reactants allows us complete control over their initial quantum degrees of freedom, whereas state-resolved, coincident detection of both products enables the probability of scattering into each of the 57 allowed rotational state-pairs to be measured. Our results show an overall agreement with a state-counting model based on statistical theory4,5,6, but also reveal several deviating state-pairs. In particular, we observe a strong suppression of population in the state-pair closest to the exoergicity limit as a result of the long-range potential inhibiting the escape of products. The completeness of our measurements provides a benchmark for quantum dynamics calculations beyond the current state of the art.

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Fig. 1: Energetics and product quantum states for ultracold reactions between KRb molecules.
Fig. 2: State-resolved coincidence detection of reaction products.
Fig. 3: Measured product state distribution and comparison to statistical theory.
Fig. 4: Influence of the long-range potential on product formation near the energy threshold.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability

The computer codes used for theoretical calculations in this study are available from H.G. ( upon reasonable request.


  1. 1.

    Li, J., Zhao, B., Xie, D. & Guo, H. Advances and new challenges to bimolecular reaction dynamics theory. J. Phys. Chem. Lett. 11, 8844–8860 (2020).

    CAS  Google Scholar 

  2. 2.

    Quéméner, G. & Julienne, P. S. Ultracold molecules under control! Chem. Rev. 112, 4949–5011 (2012).

    Google Scholar 

  3. 3.

    Croft, J. et al. Universality and chaoticity in ultracold K + KRb chemical reactions. Nat. Commun. 8, 15897 (2017).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  4. 4.

    Light, J. C. Statistical theory of bimolecular exchange reactions. Discuss. Faraday Soc. 44, 14–29 (1967).

    Google Scholar 

  5. 5.

    Nikitin, E. E. & Umanskii, S. Y. Theory of Slow Atomic Collisions Vol. 30 (Springer Science & Business Media, 2012).

  6. 6.

    Pechukas, P. in Dynamics of Molecular Collisions 269–322 (Springer, 1976).

  7. 7.

    Julienne, P. S. Ultracold molecules from ultracold atoms: a case study with the KRb molecule. Faraday Discuss. 142, 361–388 (2009).

    ADS  CAS  Google Scholar 

  8. 8.

    Balakrishnan, N. Ultracold molecules and the dawn of cold controlled chemistry. J. Chem. Phys. 145, 150901 (2016).

    ADS  CAS  Google Scholar 

  9. 9.

    Tarbutt, M. R. Laser cooling of molecules. Contemp. Phys. 59, 356–376 (2018).

    ADS  CAS  Google Scholar 

  10. 10.

    Toscano, J., Lewandowski, H. & Heazlewood, B. R. Cold and controlled chemical reaction dynamics. Phys. Chem. Chem. Phys. 22, 9180–9194 (2020).

    CAS  Google Scholar 

  11. 11.

    Ospelkaus, S. et al. Quantum-state controlled chemical reactions of ultracold potassium-rubidium molecules. Science 327, 853–857 (2010).

    ADS  CAS  Google Scholar 

  12. 12.

    Ni, K.-K. et al. Dipolar collisions of polar molecules in the quantum regime. Nature 464, 1324 (2010).

    ADS  CAS  Google Scholar 

  13. 13.

    Guo, M. et al. Dipolar collisions of ultracold ground-state bosonic molecules. Phys. Rev. X 8, 041044 (2018).

    CAS  Google Scholar 

  14. 14.

    Yang, H. et al. Observation of magnetically tunable Feshbach resonances in ultracold 23Na40K + 40K collisions. Science 363, 261–264 (2019).

    ADS  CAS  Google Scholar 

  15. 15.

    de Jongh, T. et al. Imaging the onset of the resonance regime in low-energy NO-He collisions. Science 368, 626–630 (2020).

    ADS  Google Scholar 

  16. 16.

    Rui, J. et al. Controlled state-to-state atom-exchange reaction in an ultracold atom–dimer mixture. Nat. Phys. 13, 699 (2017).

    CAS  Google Scholar 

  17. 17.

    Wolf, J. et al. State-to-state chemistry for three-body recombination in an ultracold rubidium gas. Science 358, 921–924 (2017).

    ADS  CAS  Google Scholar 

  18. 18.

    Yang, X. State-to-state dynamics of elementary bimolecular reactions. Annu. Rev. Phys. Chem. 58, 433–459 (2007).

    ADS  CAS  Google Scholar 

  19. 19.

    Clary, D. C. Theoretical studies on bimolecular reaction dynamics. Proc. Natl Acad. Sci. USA 105, 12649–12653 (2008).

    ADS  CAS  Google Scholar 

  20. 20.

    Croft, J. F. & Bohn, J. L. Long-lived complexes and chaos in ultracold molecular collisions. Phys. Rev. A 89, 012714 (2014).

    ADS  Google Scholar 

  21. 21.

    Liu, Y. et al. Photo-excitation of long-lived transient intermediates in ultracold reactions. Nat. Phys. 16, 1132–1136 (2020).

    ADS  CAS  Google Scholar 

  22. 22.

    Gregory, P. D., Blackmore, J. A., Bromley, S. L. & Cornish, S. L. Loss of ultracold 87Rb123Cs molecules via optical excitation of long-lived two-body collision complexes. Phys. Rev. Lett. 124, 163402 (2020).

    ADS  CAS  Google Scholar 

  23. 23.

    Bonnet, L. & Rayez, J. C. Some key factors of energy distributions in the products of complex-forming elementary reactions. Phys. Chem. Chem. Phys. 1, 2383–2400 (1999).

    CAS  Google Scholar 

  24. 24.

    Balucani, N. et al. Experimental and theoretical differential cross sections for the N(2D)+ H2 reaction. J. Phys. Chem. A 110, 817–829 (2006).

    CAS  Google Scholar 

  25. 25.

    Sun, Z. et al. State-to-state dynamics of H + O2 reaction, evidence for nonstatistical behavior. J. Am. Chem. Soc. 130, 14962–14963 (2008).

    CAS  Google Scholar 

  26. 26.

    Rivero-Santamaría, A. et al. The O(1D)+ H2 (X1Σ+, v, j) → OH (X2∏, v′, j′)+ H(2S) reaction at low collision energy: when a simple statistical description of the dynamics works. Phys. Chem. Chem. Phys. 13, 8136–8139 (2011).

    Google Scholar 

  27. 27.

    González-Martínez, M. L., Dulieu, O., Larrégaray, P. & Bonnet, L. Statistical product distributions for ultracold reactions in external fields. Phys. Rev. A 90, 052716 (2014).

    ADS  Google Scholar 

  28. 28.

    Nesbitt, D. J. Toward state-to-state dynamics in ultracold collisions: lessons from high-resolution spectroscopy of weakly bound molecular complexes. Chem. Rev. 112, 5062–5072 (2012).

    CAS  Google Scholar 

  29. 29.

    Ni, K.-K. et al. A high phase-space-density gas of polar molecules. Science 322, 231–235 (2008).

    ADS  CAS  Google Scholar 

  30. 30.

    Liu, Y., Grimes, D. D., Hu, M.-G. & Ni, K.-K. Probing ultracold chemistry using ion spectrometry. Phys. Chem. Chem. Phys. 22, 4861–4874 (2020).

    CAS  Google Scholar 

  31. 31.

    Hu, M.-G. et al. Direct observation of bimolecular reactions of ultracold KRb molecules. Science 366, 1111–1115 (2019).

    ADS  CAS  Google Scholar 

  32. 32.

    Hu, M.-G. et al. Nuclear spin conservation enables state-to-state control of ultracold molecular reactions. Nat. Chem. (2020).

  33. 33.

    Byrd, J. N., Montgomery, J. A. Jr & Côté, R. Structure and thermochemistry of K2Rb, KRb2, and K2Rb2. Phys. Rev. A 82, 010502 (2010).

    ADS  Google Scholar 

  34. 34.

    Yang, D. et al. A global full-dimensional potential energy surface for the K2Rb2 complex and its lifetime. J. Phys. Chem. Lett. 11, 2605–2610 (2020).

    CAS  Google Scholar 

  35. 35.

    Liu, K. Product pair correlation in bimolecular reactions. Phys. Chem. Chem. Phys. 9, 17–30 (2007).

    CAS  Google Scholar 

  36. 36.

    Brouard, M., O’Keeffe, P. & Vallance, C. Product state resolved dynamics of elementary reactions. J. Phys. Chem. A 106, 3629–3641 (2002).

    CAS  Google Scholar 

  37. 37.

    Continetti, R. E. Coincidence spectroscopy. Annu. Rev. Phys. Chem. 52, 165–192 (2001).

    ADS  CAS  Google Scholar 

  38. 38.

    Lee, S. K. et al. Coincidence ion imaging with a fast frame camera. Rev. Sci. Instrum. 85, 123303 (2014).

    ADS  Google Scholar 

  39. 39.

    Wasserman, L. All Of Statistics: A Concise Course In Statistical Inference (Springer, 2004).

  40. 40.

    Meyer, E. R. & Bohn, J. L. Product-state control of bi-alkali-metal chemical reactions. Phys. Rev. A 82, 042707 (2010).

    ADS  Google Scholar 

  41. 41.

    Amiot, C., Vergès, J. & Fellows, C. E.The long-range potential of the K2 \({X}^{1}{\Sigma }_{g}^{+}\) ground electronic state up to 15 Å. J. Chem. Phys. 103, 3350–3356 (1995).

  42. 42.

    Seto, J. Y., Le Roy, R. J., Verges, J. & Amiot, C. Direct potential fit analysis of the \({X}^{1}{\Sigma }_{g}^{+}\) state of Rb2: nothing else will do! J. Chem. Phys. 113, 3067–3076 (2000).

  43. 43.

    Yang, D., Huang, J., Hu, X., Xie, D. & Guo, H. Statistical quantum mechanical approach to diatom–diatom capture dynamics and application to ultracold KRb + KRb reaction. J. Chem. Phys. 152, 241103 (2020).

    CAS  Google Scholar 

  44. 44.

    Brumer, P., Bergmann, K. & Shapiro, M. Identical collision partners in the coherent control of bimolecular reactions. J. Chem. Phys. 113, 2053–2055 (2000).

    ADS  CAS  Google Scholar 

  45. 45.

    Kendrick, B., Hazra, J. & Balakrishnan, N. The geometric phase controls ultracold chemistry. Nat. Commun. 6, 7918 (2015).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  46. 46.

    Falke, S., Sherstov, I., Tiemann, E. & Lisdat, C. The \({A}^{1}{\Sigma }_{u}^{+}\) state of K2 up to the dissociation limit. J. Chem. Phys. 125, 224303 (2006).

  47. 47.

    Amiot, C. Laser-induced fluorescence of Rb2: the (1) \({}^{1}{\Sigma }_{g}^{+}(X)\), (2) \({}^{1}{\Sigma }_{g}^{+}\), (1) \({}^{1}{\Pi }_{u}(B)\), (1) \({}^{1}{\Pi }_{g}\), and (2) \({}^{1}{\Pi }_{g}(C)\) electronic states. J. Chem. Phys. 93, 8591–8604 (1990).

  48. 48.

    Aldegunde, J. & Hutson, J. M. Hyperfine energy levels of alkali-metal dimers: ground-state homonuclear molecules in magnetic fields. Phys. Rev. A 79, 013401 (2009).

    ADS  Google Scholar 

  49. 49.

    Caldwell, C., Engelke, F. & Hage, H. High resolution spectroscopy in supersonic nozzle beams: the Rb2 \({B}^{1}{\Pi }_{u}-{X}_{1}{\Sigma }_{g}^{+}\) band system. Chem. Phys. 54, 21–31 (1980).

  50. 50.

    Amiot, C. & Vergès, J. Optical-optical double resonance and Fourier transform spectroscopy: the Rb2 \({B}_{1}{\Pi }_{u}\) electronic state up to the quasibound energy levels. Chem. Phys. Lett. 274, 91–98 (1997).

  51. 51.

    Ullrich, J. et al. Recoil-ion and electron momentum spectroscopy: reaction-microscopes. Rep. Prog. Phys. 66, 1463 (2003).

    ADS  CAS  Google Scholar 

  52. 52.

    Vredenborg, A., Roeterdink, W. G. & Janssen, M. H. A photoelectron-photoion coincidence imaging apparatus for femtosecond time-resolved molecular dynamics with electron time-of-flight resolution of σ = 18 ps and energy resolution ΔE/E = 3.5%. Rev. Sci. Instrum. 79, 063108 (2008).

    ADS  Google Scholar 

  53. 53.

    Eppink, A. T. & Parker, D. H. Velocity map imaging of ions and electrons using electrostatic lenses: application in photoelectron and photofragment ion imaging of molecular oxygen. Rev. Sci. Instrum. 68, 3477–3484 (1997).

    ADS  CAS  Google Scholar 

  54. 54.

    Gericke, K.-H. Correlations between quantum state populations of coincident product pairs. Phys. Rev. Lett. 60, 561 (1988).

    ADS  CAS  Google Scholar 

  55. 55.

    Lin, J. J., Zhou, J., Shiu, W. & Liu, K. State-specific correlation of coincident product pairs in the F + CD4 reaction. Science 300, 966–969 (2003).

    ADS  CAS  Google Scholar 

  56. 56.

    Dixit, S., Levin, D. & McKoy, B. Resonant enhanced multiphoton ionization studies in atomic oxygen. Phys. Rev. A 37, 4220 (1988).

    ADS  CAS  Google Scholar 

  57. 57.

    Johnson, B. An improved log derivative method for inelastic scattering. J. Comput. Phys. 13, 445−449 (1973).

    ADS  Google Scholar 

  58. 58.

    Manolopoulos, D. An improved log derivative method for inelastic scattering. J. Chem. Phys. 85, 6425–6429 (1986).

    ADS  CAS  Google Scholar 

  59. 59.

    Johnson, B. A generalized JWKB approximation for multichannel scattering. Chem. Phys. 2, 381–399 (1973).

    Google Scholar 

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We thank L. Zhu for experimental assistance; T. Rosenband, G. Quéméner, W. Cairncross, E. Heller and M. Soley for discussions; T. Karman for providing the code for state-counting; J. Huang for ab initio calculations; and L. Liu for a critical reading of the manuscript. This work is supported by the DOE Young Investigator Program (DE-SC0019020) and the David and Lucile Packard Foundation. M.A.N. is supported by the Arnold O. Beckman Postdoctoral Fellowship in Chemical Instrumentation. D.Y. and D.X. acknowledge support from the National Natural Science Foundation of China (grant numbers 21733006). H.G. thanks the Army Research Office (W911NF-19-1-0283) for funding.

Author information




The experimental work and data analysis were carried out by Y.L., M.-G.H., M.A.N. and K.-K.N. Theoretical calculations were performed by D.Y., D.X. and H.G. All authors contributed to interpreting the results and writing the manuscript.

Corresponding authors

Correspondence to Yu Liu or Kang-Kuen Ni.

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Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature thanks Simon Cornish, Nandini Mukherjee and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Fig. 1 Timing diagram for product ionization.

The relative timing between the ODT, REMPI, and cleanup pulses during the state-selective ionization of reaction products. The inset shows a close-up view of a single modulation period. Unperturbed reactions occur during the dark phase of the period, while reactions influenced by the (1,064 nm) ODT light occur during the bright phase. The numbers in parentheses indicate pulse durations.

Extended Data Fig. 2 Modelling the geometric factor for product sampling.

The plot shows the fraction of product pairs that remain within the REMPI beam volume at the time of ionization as a function of the velocity of the K2 product within the pair. Blue and red dashed lines indicate the maximum velocities of the K2 and Rb2 products, respectively. The inset shows the ionization geometry.

Extended Data Fig. 3 Modelling the Doppler factor for product sampling.

a, Normalized optical intensity profiles of the REMPI laser pulses. The red trace corresponds to the 648 nm or 674 nm lights, and is proportional to \({\varOmega }_{01}\). The green trace corresponds to the 532 nm light, and is proportional to \({\varGamma }_{{\rm{ion}}}\). b, The Doppler factor FDoppler(v) versus the velocity of the K2 product. The lower and upper bounds correspond to \({\varGamma }_{{\rm{i}}{\rm{o}}{\rm{n}}}/(2{\rm{\pi }})\) at peak values of 6 MHz and 14 MHz, respectively.

Extended Data Fig. 4 Distribution of product translational energy.

The measured (blue circle) and predicted (red diamond) scattering probabilities for all allowed state-pairs are plotted versus their translational energies (T). The two sets of points are offset horizontally by 0.014 cm−1 for clarity. To aid in the identification of systematic deviations, we multiply each scattering probability by a normalized Gaussian function with a 1σ width of 0.25 cm−1, and sum them up to construct broadened distributions as shown by the blue and red curves. These curves are scaled by a factor of 0.2 for convenience.

Extended Data Table 1 Molecular dissociation energies of 40K87Rb, 40K2 and 87Rb2
Extended Data Table 2 Rotational and centrifugal distortion constants for 40K2 and 87Rb2
Extended Data Table 3 The internal energy (U) and degeneracy (\({\mathcal{D}}\)) for all measured product state-pairs
Extended Data Table 4 Various measured quantities that are used towards calculating the scattering probability into each product state-pair

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Liu, Y., Hu, MG., Nichols, M.A. et al. Precision test of statistical dynamics with state-to-state ultracold chemistry. Nature 593, 379–384 (2021).

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