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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.

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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.


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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.

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Authors and Affiliations



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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The authors declare no competing interests.

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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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