Ultracold Molecular Collisions: Quasiclassical, Semiclassical, and Classical Approaches in the Quantum Regime

Abstract

Ultracold atoms and molecules provide an unprecedented view into the role of quantum effects in chemical reactions. However, these same quantum effects make ultracold chemical reactions notoriously difficult to simulate. Contrary to conventional wisdom that ultracold systems necessitate quantum approaches, with classical mechanics offering a rough approximation at best, the past three decades have seen an explosion in classical, quasiclassical, and semiclassical approaches that yield near-exact results for ultracold systemsincluding in cases where today’s quantum methods would be computationally intractable. This review illustrates the power of these approaches with 30 years’ worth of numerical and experimental evidence. The review provides a comprehensive account of the successes of universal classical post-threshold laws for low-energy atomic and molecular scattering systems, as well as the cutting edge in quasiclassical approaches that unravel the ongoing controversy surrounding long-lived collision complex lifetimes in ultracold collisions. Special attention is placed on atomic systems where classical scaling laws hold directly at zero Kelvin, which underline the predictive power classical methods can offer even in the most extreme quantum chemical environments. The review culminates with a discussion of broader implications for classically inspired approaches to prototypically quantum chemistry in the second quantum revolution.

1. Introduction

In 2012, when Chemical Reviews published its Special Issue on ultracold molecules, − the field of ultracold chemistry was at its outset. The Special Issue placed a special focus on the preparation of molecules in the cold regime (below 1 K) − and the ultracold regime (below 1 mK). − Provisions were made for future applications for condensed matter theory and aspirations toward quantum chemistry. At the time, evidence of ultracold chemical reactions was restricted to reactant lossas in the groundbreaking measurement of molecular density decay in the ultracold ⁴⁰K⁸⁷Rb dimer reaction at just 300 nK by Ospelkaus, Ni et al. In the intervening years, there has been a “quantum leap” in ultracold chemistry.

A surprising outcome of such recent developments in ultracold chemistry − is classical, semiclassical, and quasiclassical mechanics is sufficient to simulate aspects of a wide range of cold and ultracold chemical reactions nearly exactly. This advancement provides an opportunity to develop computational tools that vastly reduce the cost of cold and ultracold simulations and aid interpretation of fundamental nonreactive and reactive collision processes. Section provides a comprehensive account of the numerical and experimental evidence in the past 25 years, including the ubiquitous ability of classical and semiclassical mechanics to closely reproduce quantum cross sections for atom–ion collisions at collision energies as low as 10 nK, the capability of classical Langevin capture theory to yield accurate reactive rate coefficients for triatomic collisions at collision energies as low as 10 nK, and the power of semiclassical mechanics to evidence a quasiuniversal quantum-to-classical crossover in dipolar collisions at collision energies as low as 7 pK. Note the lowest of these temperatures is 12 orders of magnitude colder than the interstellar medium. These findings go beyond theoretical results. Classical and semiclassical mechanics have been seen to closely agree, for example, with experimentally obtained cross sections on the order of 10 μK for Li + Ba⁺ collisions, 10 mK for H₂ + HD collisions, and 0.1 K for H₂ + He collisionsthe latter with a relative error of just 1%.

Recent rationales explain how semiclassical and classical mechanics can retain such accuracy in such a counterintuitive regime. Competing theories are detailed in Section , beginning with the foundational work of Gribakin and Flambaum, who found that semiclassical cross sections can hold when not a single oscillation of the scattering wave function fits in the potential well. − This argument is compared with the semiclassical Wentzel-Kramers-Brillouin (WKB) criterion furthered by Julienne and Mies in 1989, Côté et al. in 1996, and Mody et al. in 2001 to pinpoint the onset of the quantum effects of quantum reflection and quantum suppression. Ultracold inferno theory, introduced in 2018, is also presented as a conceptual framework to indicate when not just semiclassical, but also classical treatment is expected to hold: namely, in the absence of quantum reflection and quantum interference. , These methods are placed in light of the controversy surrounding the true applicability of the WKB criterion and the related quantality condition and the question of whether these points of reference are sufficient to determine the “source” of quantum reflection effects. ,

The observation and justification of classical, semiclassical, and quasiclassical treatment of cold and ultracold collisions opens the door wide to applications throughout the field of chemistry. Section details how such advancements enable the derivation of novel semiclassical scaling laws for cross sections and rate coefficients, such as the semiclassical eikonal scaling laws of Bohn et al. for dipolar collisions currently underway in the laboratory, , as well as the development of techniques to rigorously extend existing quantum and classical cross section scaling laws to cover the whole of the collision energy region, as in the quantum threshold model of Quéméner and Bohn − and the quantum Langevin model of Gao. − Additionally, the review generalizes the semiclassical analogue of the Fermi golden rule approach of Mody et al. to derive semiclassical laws that apply to two-body collisions in not just three-dimensional, but reduced-dimensional collisions relevant to reactions in optical traps. ,− Section covers the novel perspective that rationales for classical and semiclassical behaviors in ultracold collisions offer on the origin of quantum-classical correspondence in scaling laws directly at zero collision energy in the renowned three-dimensional Coulomb potential and Wannier double ionization system, with implications for multiple ionization − and atomic fragmentation. , Section collects these universal scaling lawsspanning one- to N-dimensions and short-range to long-range potentialsto highlight and facilitate prediction of where classical and semiclassical predictions are expected to hold in a yet wider range of cold and ultracold collisions.

Section considers “sticky collisions”the mysterious disappearance of reactants even in ultracold collisions with no energetically accessible products. − Here a quasiclassical strategy based on the Rice–Ramsperger–Kassel–Marcus (RRKM) theory of unimolecular dissociation rates − has emerged as the method of choice. − This method has succeeded not least of all because simulation of such collision complexes with exact quantum mechanics ranges from highly computationally intensive for three-atom systems to computationally intractable for just four atoms. There remains a conundrumwhereas the quasiclassical and experimental collision-complex lifetimes for the ⁴⁰K ⁸⁷Rb + ⁴⁰K ⁸⁷Rb collision differ by a factor of just 2, , quasiclassical and experimental collision-complex lifetimes for ⁴⁰K⁸⁷Rb + ⁸⁷Rb differ by 5 orders of magnitude ,, namely, a key area for further study.

Section provides an outlook for the future of classical, semiclassical, and quasiclassical methods in cold and ultracold molecules. The ability of classical mechanics to reap near-exact results opens the door to simulations of systems that are computationally intractable today, which includes systems of four atoms. Rigorous criteria for quantum-classical correspondence facilitate identification of not only where classical mechanics applies, but also where quantum treatment is essential, which is especially relevant today given the importance of accurate simulation of ultracold molecules proposed for quantum sensing − and quantum computing − technologies central to the second quantum revolution. Results at cold and ultracold temperatures also have implications for room-temperature chemistrythe exponential improvement in computational cost of classical simulations relative to quantum calculations provides a valuable complement to state-of-the-art data-science approaches to quantum mechanics, as well as the means to readily visualize detailed reaction mechanisms in terms of classical particles. And, the insight these findings offer into the border between quantum and classical worlds has implications for the wider field of near-threshold phenomena throughout chemistry and physics. We have moved far beyond the ultracold chemistry of 25 years ago, and this review provides a vision of what may be in store for the next 25 years.

2. Universal Classical and Semiclassical Scaling Laws in Ultracold Molecular Systems

2.1. Evidence of Quantum-to-Classical Crossover in the Cold and Ultracold Regime

To begin, some of the strongest support for classical treatment of ultracold collisions comes in the form of universal classical and semiclassical scalings near threshold in cold and ultracold scattering cross sections. ,− ,− Modern scattering calculations ,− ,− ,− ,− and experiments − ,,,,− confirm the prototypical picture of low-energy atomic and molecular cross sections: In the absence of resonances and bound states, − the Bethe–Wigner quantum threshold law , operates at the threshold of energetic accessibility (see Table ). , As the energy increases, a quantum-to-classical crossover appears. And, at the highest energies, the cross section asymptotically obeys (semi)­classical post-threshold laws such as the classical capture theory laws of Langevin and Gorin , (modulo oscillations and resonances, see Table ). A schematic of this behavior is given in Figure .

1. Wigner Threshold Laws for Universal Scalings of Collision Cross Sections σ­(k) as a Function of the Wavenumber k for Two-Body Collisions in Three Dimensions .

	Inelastic
lim r→∞ V(r)	Endothermic	Exothermic	Depolarization	Elastic
0	$k^{2\mathcal{l}+1}$	$k^{2\mathcal{l}-1}$	$k^{2\mathcal{l}+2\mathcal{l}\prime }$	$k^{4\mathcal{l}}$
Ze2/r	exp(−2πμZe 2/(ℏ2 k))	k–2 exp(−2πμZe 2/(ℏ2 k))	k–2 exp(−4πμZe 2/(ℏ2 k))	k –2
–Ze 2/r	1	k –2	k –2	k –2

^aScalings are shown for elastic collisions and inelastic collisions with positive threshold energy (endothermic reactions), negative threshold energy (exothermic reactions), and changes in angular momentum (depolarization events) and are given for short-range (lim _r→∞ V(r)) and long-range (attractive Coulombic lim _r→∞ V(r) = −Ze ²/r and repulsive Coulombic lim _r→∞ V(r) = −Ze ²/r) potentials. Expressions are provided in terms of the reduced mass μ, charge number product Z, and initial $\mathcal{l}$ and final angular momentum ${\mathcal{l}}^{\prime }$ quantum numbers of the collision partners, where e is the electron charge and ℏ is reduced Planck’s constant.

2. Summary of Classical Langevin and Gorin , Laws for Scalings of the Capture Cross Sections σ _L (E) with the Collision Energy E for van der Waals Potentials C _n r ^–n of Order n .

n	3	4	6
σ L (E)	$3\pi {\left(\frac{C_3}{2E}\right)}^{2/3}$	$2\pi {\left(\frac{C_4}{E}\right)}^{1/2}$	$\frac{3\pi }{2}{\left(\frac{2C_6}{E}\right)}^{1/3}$

^aThe universal form of the scattering cross section, including its constant multiplicative factor, at low energy is provided. Reproduced table with permission from Robin Côté and Ionel Simbotin, Physical Review Letters, 121, 17, 173401, 2018. Copyright 2018 by the American Physical Society.

1.

Prototypical framework for near-threshold cross sections as exemplified by numerical calculation of the transmission probability P in a one-dimensional V(x) ∝ – x ^–6 potential as a function of the wavenumber k. The transmission probability follows the quantum threshold law P ∝ k at threshold (k → 0) and crosses over (k ≈ 1) to the classical post-threshold law P = 1 at higher wavenumber (k → 1).

Numerical and experimental results show the quantum-to-classical crossover frequently appears in the cold to ultracold regime in molecular collisions. ,− ,,,− ,− Such low-energy quantum-to-classical crossovers imply that classical post-threshold laws for cross section and rate coefficient scalings are also highly accurate at extremely low collision energies. Several of the lowest-energy numerical examples of ultracold quantum-to-classical crossover and operation of classical post-threshold laws in the cold and ultracold regime are depicted in Figures –, with experimental counterparts in Figures – and additional details and related work below.

2.

Numerical evidence of quantum-to-classical crossover at ultracold energies approximately on the order of 10 nK in the averaged diffusion cross section ⟨σ _d ⟩ (black line) and diffusion coefficient D (red line) for Na + Na⁺ collisions. Reprinted from Advances in Atomic, Molecular, and Optical Physics, 65, R. Côté, Ultracold atom–ion collisions, 67, Copyright 2016, with permission from Elsevier, based on reproduced figures with permission from R. Côté and A. Dalgarno, Physical Review A, 62, 1, 012709, 2000. Copyright 2000 by the American Physical Society.

7.

Comparison of the reactive (blue) and elastic (black) rate coefficients to the classical Langevin rate coefficient (dashed-dotted red line) for D⁺ + H₂ (v = 0, j = 0) indicates the reactive rate coefficient obeys the classical Langevin scaling down to the ultracold regime on the order of 10 nK. Reproduced from M. Lara and P. G. Jambrina, Cold and ultracold dynamics of the barrierless D⁺ + H₂ reaction: Quantum reactive calculations for ∼R ^–4 long-range interaction potentials, Journal of Chemical Physics, 143 (2015) 204305, with the permission of AIP Publishing.

8.

Agreement between experimental total cross section measured at 30 μK (circles) with the classical Langevin (black line), scaled classical Langevin (thin dashed lines), quantum (colorful solid lines), and quantum thermalized (thick dashed lines) cross sections for (left) Li­(2S _1/2) + Ba⁺ (5D _5/2) and (right) Li­(2S _1/2) + Ba⁺ (5D _3/2). Quantum results are shown for competing nonradiative charge exchange (NRCE, yellow), nonradiative quenching (NRQ, red), and fine-structure quenching (FSQ, green) processes. Reproduced with permission from ref by Xing et al. CC BY 2024 IOP Publishing.

13.

Experimental evidence of cold quantum-to-classical crossover in He­(2³ P ₂)/para-H₂ (j = 0) collisions. The measured reaction rate constant (black points) obeys the classical Langevin post-threshold law (red dashed line) at collision energies as low as 0.8 K, with a quantum-to-classical crossover at energies on the order of 1 K for HD (j = 0) (gray points), compared to the p-wave barrier height (vertical dashed line). Reproduced figure with permission from Y. Shagam, A. Klein, W. Skomorowski, R. Yun, V. Akerbukh, C. P. Koch, and E. Narevicius, Molecular hydrogen interacts more strongly when rotationally excited at low temperatures leading to faster reactions, Nature Chemistry, 7, 921–926, 2015, Springer Nature. Reproduced with permission from SNCSC.

2.1.1. Numerical Evidence

2.1.1.1. Atom–Ion Collisions

A number of authors have numerically identified quantum-to-classical crossover in hybrid atom–ion systems at collision energies well below 1 K. ,− ,,,, In 2000, Côté and Dalgarno identified a quantum-to-classical crossover at extremely low energies, as low as 10 nK, for collisions of Na + Na⁺. , As shown in Figures and , computed cross sections indicated that semiclassical treatment accurately modeled the quantum results at energies as low as 10^–12 au (300 nK) for elastic and charge-transfer cross sections, with classical treatment accurately reproducing quantum results at energies as low as 10 nK for charge exchange cross sections. Côté and Dalgarno determined cross sections for ab initio potential energy surfaces ${{}^2\mathrm{\Sigma }}_g^{+}$ and ${{}^2\mathrm{\Sigma }}_u^{+}{\mathrm{Na}}_2^{+}$ , with the potential approximated at short distances by an exponential wall and the potential approximated at long distances as V _g,u (R) ∼ V _disp (R) ∓ V _exch (R) for dispersion term V _disp (R) and exchange term V _disp (R). Quantum scattering cross sections were determined from phase shifts calculated from the regular solutions of the corresponding partial wave equation. The quantum scattering cross sections were compared to both the classical Langevin formula and Massey and Mohr’s high angular momentum limit of Jeffrey’s semiclassical approximation. − Côté and Dalgarno recognized the underlying calculation strategies rest only on the neutral atom’s polarizability and mass, and thus are equally applicable to other alkali-metal hybrid atom–ion systemsand therefore low-energy quantum-to-classical crossover in atom–ion systems is expected to be widespread.

3.

Numerical evidence of quantum-to-classical crossover at ultracold energies approximately on the order of 10^–12 au (300 nK) in (a) the elastic cross section of the ${{}^2\mathrm{\Sigma }}_g^{+}$ state (blue line) and ${{}^2\mathrm{\Sigma }}_u^{+}$ state (red line, semiclassical formula shown in black) and (b) the charge exchange cross section (black line, Langevin σ _L shown in green, σ _L /2 shown in blue, and higher-energy prediction in red) for Na + Na⁺ collisions. Reproduced from Advances in Atomic, Molecular, and Optical Physics, 65, R. Côté, Ultracold atom–ion collisions, 67, Copyright 2016, with permission from Elsevier, based on reproduced figures with permission from R. Côté and A. Dalgarno, Physical Review A, 62, 1, 012709, 2000. Copyright 2000 by the American Physical Society.

In 2001, Siska supported this finding, similarly identifying a quantum-to-classical crossover by collision energies of 10 μK in spin–orbit relaxation of Ne⁺ + He. Calculations were performed on an empirical potential energy surface based on spectroscopic measurements for the molecular ion and ab initio theory. Siska recognized that the required potential energy surface shares the form of an atom–homonuclear model potential, which facilitates rigorous quantum treatment according to the formalism of Reid, Arthurs, and Dalgarno. − The resulting coupled radial equations were then integrated numerically according to a variable step-size modified log-derivative method specifically for ⁴He + ²⁰Ne⁺. Although the magnitude of the cross section usually remained <0.1% of the magnitude predicted by Langevin for the polarizability considered, the quantum scaling was seen to closely agree with the classical Langevin scaling down to collision energies in the quantum Wigner threshold regime less than 10 μK, well within the ultracold regime.

Likewise, in 2020, Pandey et al. demonstrated that total cross sections for the ⁷Li⁺ – ⁷Li collision agree with semiclassical and classical scalings at collision energies just above the ultracold regime. Pandey et al. developed ab initio potential energy curves for the Li₂ electronic ground state X ²Σ _g and first-excited state A ²Σ _g according to the multireference configuration interaction (MRCI) method, which were used to determine the cross section quantum mechanically via direct numerical solution of the radial Schrödinger equation for individual partial waves and semiclassically according to the work of Côté and Dalgarno. The scattering cross sections for ${}^7\mathrm{L}{\mathrm{i}}_2^{+}$ for all computed potential energy curves agreed with the semiclassical, classical Langevin, or scaled classical Langevin cross sections at collision energies at or greater than on the order of 10 mK.

In line with these results, in 2009, Zhang et al. identified quantum-to-classical crossover at 10 mK to 1 μK for inelastic and elastic ⁶Li + ⁷Li⁺ → ⁶Li⁺ + ⁷Li collisions, respectively. Zhang et al. considered cross sections computed for Born–Oppenheimer potentials with nonadiabatic couplings via the multiconfiguration self-consistent field (MCSCF) theory and multireference configuration interaction (MRCI) calculations. A comparison of quantum scattering cross sections and the semiclassical Langevin charge exchange model of Côté and Dalgarno , again demonstrated quantum-to-classical crossover from Wigner’s quantum threshold law beginning at 10^–10 eV (1 μK), with semiclassical mechanics yielding accurate elastic scattering cross sections at energies higher than 10^–6 eV (10 mK).

And, in 2011, Zhang et al. further showed that semiclassical mechanics reproduces quantum scattering cross sections at lower energy for charge exchange in atom–ion Be₂ collisions. Zhang et al. again calculated ab initio quantum scattering cross sections on Born–Oppenheimer potentials with nonadiabatic couplings and compared the results to Côté and Dalgarno’s semiclassical Langevin charge model. Here, quantum-to-classical crossovers were identified in these results at 0.1 K to 0.1 mK.

Côté and Simbotin further extended the use of semiclassical mechanics to accurately predict ultracold atom–ion collision cross sections to temperatures up to 10 orders of magnitude colder (just above fK) in 2018, as shown in Figure . Côté and Simbotin published findings on quantum-to-classical crossover in the ultracold regime for resonant charge transfer of Yb + Yb⁺, spin flip of Na + Ca⁺ and ⁸⁵Rb + ⁸⁷Rb, and excitation exchange of ${}^{133}\mathrm{C}{\mathrm{s}}^{+}+{}^{133}\mathrm{C}\mathrm{s}(6p)$ – the last well below 1 pK. For Yb₂ , quantum results were considered for V _{g /u} potentials’ ${{}^2\mathrm{\Sigma }}_g^{+}$ and ${{}^2\mathrm{\Sigma }}_u^{-}$ states; for Na + Ca⁺, singlet A ¹Σ⁺ and triplet a ³Σ⁺ states; , for ⁸⁵Rb + ⁸⁷Rb, for singlet X ¹Σ _g and triplet a ³Σ _u states; and for Cs⁺ + Cs* (6p), for two Σ _g/u and two Π _g/u states. , van der Waals potentials of order four, four, six, and three approximated the asymptotic potentials, respectively. Semiclassical Wentzel–Kramers–Brillouin (WKB) treatment − of the cross section for third-, fourth-, and sixth-order van der Waals potentials were directly related to the classical Langevin cross section to the expression for s-wave scattering; which yielded a modified Langevin scaling law for excitation exchange that incorporates the impact of quantum s-wave scattering above the prototypical Wigner quantum regime. This semiclassical modified Langevin scaling was again found to hold above a quantum-to-classical crossover that appears at energies on the order of approximately sub-mK to mK for spin-flip collisions of ^A Rb + ^B Rb for A = 85, 87 and B = 85, 87; μK to nK for resonant charge transfer in ^A Yb + ^A Yb⁺ for A = 168, 170, 172, 173, 174, 176; μK to nK for spin-flip collisions of Na + ^A Ca⁺ for A = 40, 42, 43, 44; and pK to fK in excitation exchange in ^A Cs + ^A Cs⁺ for A = 133, 132.75. These results, which encompass hybrid atom–ion systems for a variety of species of differing isotopes, both predict that quantum s-wave signatures are accessible experimentally via measurement of the semiclassical modified Langevin cross section and fulfill Côté and Dalgarno’s projection two decades prior , that their semiclassical treatment again applies broadly to alkali-metal hybrid atom–ion systems.

4.

Numerical example in which quantum-to-classical crossover appears at energies between on the order of 1 fK and 1 pK in excitation exchange in (a) ${}^{133}\mathrm{C}{\mathrm{s}}^{+}+{}^{133}\mathrm{C}\mathrm{s}(6p)$ and (b) in the same system with a rescaled fictitious mass m _Cs = 132.75 that models an isotope effect. Images directly compare quantum numerical results (solid black line) with the classical Langevin cross section σ _L (blue dot-dashed line), s-wave contribution $\frac{\pi }{k^2}{\sin }^2\mathrm{\Delta }{\delta }_0$ (black dashed line), Langevin cross section s-wave contribution σ _L sin²Δδ₀ (solid blue line), and the s-wave-scattering-modulated Langevin cross section ${\sigma }_{\mathrm{exc}}(E)$ = $[\frac{\pi }{k^2}+{\sigma }_L(E)]{\sin }^2\mathrm{\Delta }{\delta }_0(E)$ . Reproduced figure with permission from Robin Côté and Ionel Simbotin, Physical Review Letters, 121, 17, 173401, 2018. Copyright 2018 by the American Physical Society.

In 2021, Hirzler and Pérez-Ríos demonstrated that quasiclassical and classical mechanics is also sufficient to accurately simulate another type of atom–ion collision processcharge exchange cross sections for low-energy Rydberg atom–ion collisions of Li* – Li⁺ and Li* – Cs⁺. Hirzler and Pérez-Ríos considered the interaction between the ionic cores and the Rydberg electron in resonant charge exchange according to two models: (i) a Coulomb potential and (ii) a pseudopotential that consists of a screened Coulomb interaction term and Rydberg electron–core electron interaction term. Scattering cross sections were determined quasiclassically according to a quasiclassical trajectory (QCT) approach ,− that treated nuclear dynamics classically and related the trajectories’ initial conditions and final positions to individual quantum states semiclassically. In order to examine the regions of parameter space well described by QCT and classical Langevin scaling, the number of partial waves required for accurate simulation of cross sections was determined, with the result that QCT remains accurate to energies as low as on the order of 1 K, and a classical Langevin framework holds to energies as low as on the order of that associated with Bose-Einstein condensation, namely, 100 nK.

2.1.1.2. Molecule–Molecule Collisions

In 2009, Bohn, Cavagnero, and Ticknor found quasiuniversal quantum-to-classical crossover behavior also holds for dipolar collisions in the ultracold regime, of particular interest today given ongoing ultracold dipolar collision experiments. − ,,− In order to generate a widely applicable formulation for scattering cross sections, Bohn, Cavagnero, and Ticknor considered a potential dictated solely by two-body dipole–dipole interactions, with short-range physics and internal structure of the dipole neglected, as supported by previous work by Ticknor. , The resulting scattering cross section thus encompassed dipolar collisions in general, with the identity of the molecules of interest imparting on the resulting scattering cross sections only a length scale D = M⟨μ₁⟩⟨μ₂⟩/ℏ² and energy scale E _D = ℏ⁶/M ³ ⟨μ₁⟩² ⟨μ₂⟩², where M is the reduced mass and μ₁ and μ₂ are the dipoles. (Note that since both scales are dependent on the dipole moment, they are also dependent on the strength of any applied electric field.) Close-coupling results for this dipolar collision system were compared to the quantum Born approximation and the semiclassical eikonal approximation.

Semiclassical mechanics was found to be accurate throughout the cold collision regime to energies on the order of the energy scale E _D , usually nK to μK. A quantum-to-classical crossover is therefore expected at 445 nK for OH, 83 nK for KRb, and 7 pK for LiCs for fully polarized molecules, as shown in Figure well within the ultracold regime. Bohn, Cavagnero, and Ticknor additionally recognized that in the quantum-to-classical crossover regime, universality falters and the cross section rests on lengths shorter than the length scale D, such that examination of the relevant energy regime may facilitate exploration of short-range physics and the intermolecular potential energy surface.

5.

Quantum-to-classical crossover of the total scattering cross section (black line) from the quantum Born approximation scaling (blue line) to the semiclassical eikonal approximation scaling (red line) occurs at characteristic energy E _D = ℏ⁶/M ³ ⟨μ₁⟩² ⟨μ₂⟩² in distinguishable dipolar molecules, which corresponds to ultracold temperatures of 7 pK for LiCs, 83 nK in KRb, and 445 pK in OH. Reproduced with permission from ref by Bohn, Cavagnero, and Ticknor. CC BY 2009 IOP Publishing.

Dashevskaya et al. likewise showed that complex formation rates for a wide range of diatomic-diatomic/polyatomic ion collisions are amenable to quasiclassical treatment at temperatures above on the order of 1 to 10 mK in 2005and further showed that this behavior is expected to be widespread for molecular ion-neutral ion collisions. Dashevskaya et al. considered an asymptotic diatomic-ion interaction potential given by a sum of terms for an isotropic charge-induced dipole and an anisotropic charge-induced dipole with a charge-induced quadrupole. The total capture rate coefficient was determined in the zero-energy limit according to the quantum Bethe expression of Eltschka, Moritz, and Friedrich; in the intermediate energy range by direct solution of quantum coupled equations and the adiabatic channel classical (ACCl) approximation; , and in the high-energy limit by the classical Langevin rate. Quasiuniversal behavior of the capture cross section was observed, as shown in Figure , in which the total capture rate coefficient transitioned to its classical Langevin value at temperatures above T _VLT, the characteristic temperature (termed Very Low Temperature) at which the de Broglie wavelength of relative motion equals the Langevin capture radius. This characteristic temperature was found to appear at T _VLT = 1.49 · 10^–4 K for collisions of diatomic nitrogen N₂ with an ion X⁺ of mass 10 amu, and in the range 1.43 to 17.2 mK for a wide array of collisions of diatomic hydrogen isotopologues with molecular ions, as shown in Table . In 2016, Dashevskaya et al. confirmed accurate quantum capture rate coefficients for H₂ (j = 0, 1) + H₂ collisions approach the classical Langevin value on the order of 1 to 10 mK. These findings point to a broad ability of quasiclassical simulations to accurately model cold collisions of diatomic-molecular ion collisions in the cold regime below 1 K.

6.

Comparison of the total capture rate coefficient scaled to the classical Langevin rate for H₂ (j = 0) collisions with any ionic partner indicate close agreement between the quantum (χ _j=0 = k _j=0/k ^L, solid line), ACCl (Χ _j=0 , dashed line), quantum mechanically amplified ACCl (open circles), and Langevin rates (1.0) above the characteristic temperature T_VLT. See ref for additional details. Adapted from E. I. Dashevskaya and I. Litvin, Rates of complex formation in collisions of rotationally excited homonuclear diatoms with ions at very low temperatures: Application to hydrogen isotopes and hydrogen-containing ions, Journal of Chemical Physics, 122 (2005) 184311, with the permission of AIP Publishing.

3. Characteristic Temperature T _VLT for Transition to the Classical Langevin Capture Rate in Collisions of Diatomic Hydrogen Isotopologues with Selected Diatomic and Polyatomic Isotopologues .

Molecular Ion	Neutral Molecule	TVLT [mK]
H2	H2	17.2
	HD	12.0
	D2	9.70
HD+	H2	12.0
	HD	7.65
	D2	5.86
D2	H2	9.70
	HD	5.86
	D2	4.30
H3	H2	12.0
	HD	7.65
	D2	5.86
H2D+	H2	9.70
	HD	5.86
	D2	4.30
HD2	H2	8.43
	HD	4.90
	D2	3.48
D3	H2	7.65
	HD	4.30
	D2	3.00
CH3	H2	5.53
	HD	2.75
	D2	1.73
C2H2	H2	5.00
	HD	2.38
	D2	1.43

^aNote all temperatures considered occur in the cold regime well below 1 K. Adapted from E. I. Dashevskaya and I. Litvin, Rates of complex formation in collisions of rotationally excited homonuclear diatoms with ions at very low temperatures: Application to hydrogen isotopes and hydrogen-containing ions, Journal of Chemical Physics, 122 (2005) 184311, with the permission of AIP Publishing.

2.1.1.3. Atom–Molecule and Ion–Molecule Collisions

Likewise, numerous authors have identified ultracold quantum-to-classical crossover in triatomic collisions, including those relevant to experiments on three-body recombination (see refs , , , , , , , , , − , − ). In 2015, Lara et al. recognized that the classical Langevin scaling law accurately yields the reactive rate coefficient in the triatomic D⁺ + H₂ (v = 0, j = 0) reaction in the ultracold regime as low as 10 nK, see Figure . Here, quantum calculations were performed according to the quantum reactive scattering method of Launey and Dourneuf on the potential energy surfaces of Velilla et al. and Aguado et al. Classical calculations were performed according to the classical Langevin model for long-range fourth-order van der Waals potentials both in its standard form and according to a numerical-capture statistical model that incorporates numerical values for centrifugal barrier heights to capture short-range behaviors, discrete summation over partial wave components, and a statistical factor to model the probability a collision complex decomposes into products. In its standard form, the classical Langevin model agrees with the quantum Wigner threshold law for long-range fourth-order van der Waals potentials, such that the classical Langevin model was found to accurately predict the E ⁰ scaling of the reactive cross section for D⁺ + H₂ (v = 0, j = 0) collisions both in the low-energy and high-energy limits. Moreover, not only do the scalings agree, but the values of the cross sections also agree for this collision system (with the quantum result only 10% larger than the classical value), such that the cross section is predicted to vary by a multiplicative factor of just <10 from the classical value throughout the ultracold regime and extending beyond 10 orders of magnitude in energy.

In 2024, Karpa and Dulieu underlined the ubiquity of classical behavior in ultracold ion–molecule collisions according to a semiclassical Langevin scattering model. , The characteristic length and energy scales of s-wave collisions, defined by the position R ^* of the centrifugal barrier maximum E ^* associated with p-wave collisions $\mathcal{l}=1$ in the absence of a Langer correction, were considered for an ion-neutral system with reduced mass μ and fourth-order van der Waals coefficient C ₄

$$R^{*}=\sqrt{2\mu C_4/{\hslash }^2}$$ 1

$$E^{*}=\frac{{\hslash }^4}{4{\mu }^2{C}_4}=\frac{{\hslash }^2}{2\mu {(R^{*})}^2}$$ 2

With the permanent electric dipole moments and static dipole polarizabilities of Vexiau et al., the characteristic scales were computed for collisions of ¹³⁸Ba⁺ with bialkali molecules currently of interest in ultracold molecular experiments, revealing characteristic energies on the order of pK to nK, as shown in Table . Karpa and Dulieu suggested that this implies that even at the coldest temperatures at which molecular experiments have been performed as of 2024 (on the order of 10 nK), − ion–molecule collisions involving the species in Table would behave according to the classical Langevin capture model.

4. Characteristic Semiclassical Langevin Energy E ^* and Length R ^* Scales of the s-Wave Regime in ¹³⁸Ba⁺ Ion-Neutral Rovibronic Ground-State Molecule Collisions, with Ion–Atom Collisions for Reference .

Molecule	R* [10–6 m]	E* [nK]
LiCs	25.9	0.00521
NaCs	40.1	0.00206
LiRb	16.2	0.0165
LiK	10.0	0.0703
NaRb	23.6	0.00711
NaK	14.2	0.0281
KCs	22.2	0.00644
RbCs	21.6	0.00614
KRb	6.21	0.0952
LiNa	1.12	7.86
Rb	0.295	52.3
Li	0.0694	8760.

^aFor all systems considered, the characteristic energy falls in the ultracold regime. Reproduced with permission from ref by Karpa and Dulieu. CC BY 2025 arXiv.

A great number of numerical studies have found classical, semiclassical, and quasiclassical mechanics are sufficient to accurately predict many triatomic scattering cross sections at energies above 0.1 mK in triatomic collisions (see refs , , , , , , , , , − , ). In 2000, Dashevskaya and Nikitin demonstrated that quasiclassical mechanics can be used to accurately determine s-wave vibrational relaxation cross sections for H₂ (ν = 1, j = 0) + ⁴He → H₂ (ν = 0, j = 0) + ⁴He in the ultracold regime above 0.1 mK. Dashevskaya and Nikitin expanded the Landau–Lifshitz method to determine quasiclassical matrix elements , to apply to states that are quasiclassical only in that their asymptotics obey the WKB approximation. This generalized Landau–Lifshitz method was applied to an exponential repulsive interaction and a Morse potential model and the results were compared to close-coupling cross sections. Quasiclassical results for the exponential repulsive interaction and for the Morse interaction closely agreed with quantum results just above 10^–8 eV (0.1 mK) with the inclusion of quantum suppression effects. Dashevskaya et al. subsequently identified quantum-to-classical crossover in ion–molecule capture of H₂ (j = 1) + Ar⁺ on the order of 0.1 mK and N₂ (j = 1) + He⁺ on the order of 1 mK in 2004. Dashevskaya et al. treated the ion–molecule capture process in terms of a closed-state, structureless atomic ion and a rigid rotor molecule and found close agreement between rate coefficients computed with accurate quantum mechanics and the adiabatic channel classical approximation. , This agreement allowed for the accurate analysis of the capture process via classical treatment of the relative motion of the collision partners under an assumption of conserved intrinsic angular momentum projected onto the collision axis.

Similarly, in 2005, Quéméner et al. found that quenching rates in ultracold K + K₂ collisions obey the classical Langevin model at energies higher than 0.1 mK. , An ab initio potential energy surface was generated for the potassium trimer 1⁴ A ₂ state via single-reference restricted open-shell coupled-cluster with single, double, and noniterative triple excitations [RCCSD­(T)] − and employed the resulting surface in quantum close-coupling calculations with a body-frame democratic hyperspherical coordinate formalism at short atom–diatom distances , and the Arthurs–Dalgarno formalism at long atom–diatom distances. The resulting quantum total quenching rate coefficients reached their classical Langevin capture model values semiquantitatively past 0.1 mK. In 2017, Croft et al. also calculated elastic cross sections for s-wave K₂ (v = 0, j = 0) + Rb collisions that exhibited a quantum-to-classical crossover on the order of 0.1 mK. Croft et al. evaluated quantum cross sections according to the atom–diatom scattering method of Pack et al. , and observed a large number of fine resonances appearing above approximately 10 mK. Although not explicitly discussed with reference to the classical post-threshold law in the work, resonances are visible beyond the energy at which the cross section begins to deviate from the threshold regime associated with E ⁰ quantum scaling in the energy range beyond 0.1 mK.

And, likewise, in 2016, Stoecklin et al. found that quantum vibrational quenching cross sections agree with the classical Langevin capture scaling at collision energies above 1 mK for the collision of Ca and BaCl⁺. Stoecklin et al. created an analytic potential energy surface for the electronic ground state ¹ A′ via ab initio calculation at the coupled cluster singles, doubles, and perturbative triples [CCSD­(T)] level of theory and employed both the close-coupling method and quantum defect theory − to determine the cross section quantum mechanically. The classical Langevin rate was computed and used to determine a statistical capture rate constant according to the method of Lara et al. Additionally, results were compared to the experimental data of Campbell et al. on the order of 10 to 100 mK. Both the experimental and quantum close-coupling vibrational quenching rate coefficients were found to obey the classical Langevin scaling on the order of 10 to 100 mK. Likewise, quantum defect theory universal loss rate coefficients were found to agree closely with the classical Langevin law (modulo oscillations) at energies on the order of 10 nK and above, and despite discrepancies at energies below one mK, was consistent with the experimental data and close-coupling calculations at energies above one mK.

Additional studies identify triatomic collisions in which classical post-threshold laws operate at collision energies above 10 mK. ,, In 2005, Cvitaš et al. found that classical post-threshold laws hold for the total inelastic rate coefficient of Li + Li₂ collisions above 10 mK. Cvitaš et al. considered an ab initio global potential energy surface for Li₃ computed via all-electron coupled-cluster theory. To determine scattering cross sections, the scattering S matrix was determined through wave function matching at the boundary of an inner region, where the wave function is determined through propagation of coupled equations in a diabatic-by-sector basis, and an outer region, where the wave function is determined according to the Arthurs–Dalgarno formalism. Comparison of the resulting quenching rates to the classical Langevin rate coefficient for vibrational quantum numbers v = 1, 2 indicated semiquantitative agreement for bosonic and fermionic Li. It was noted that the same Langevin rate coefficient holds for other spin-polarized state alkali atom–diatom inelastic rates in the post-threshold regime, which Cvitaš et al. predict to occur at or above 6.86 mK for Na, 1.89 mK for K, 0.537 mK for Rb, and 0.235 mK for Cs. In a similar spirit, in 2023, Morita et al. demonstrated that numerical quantum reactive and elastic cross sections obey classical post-threshold laws above 10 mK for Li + NaLi­(ν = 0, j = 0) → Li₂ + Na. Morita et al. considered an ab initio potential energy surface for the 1⁴ A′ state of NaLi₂ calculated according to the multireference configuration interaction (MRCI) method , and performed accurate quantum reactive scattering calculations with the adiabatically adjusting principal-axis frame hyperspherical (APH) coordinates method of Pack et al. , Examination of the reactive and elastic cross sections as a function of the collision energy evidence a rate law change consistent with quantum-to-classical crossover on the order of 10 mK and below for quantum numbers J = 0, 1 for both even and odd parity.

Beyond classical and semiclassical mechanics, beginning in 2019 Pérez-Ríos et al. also identified that QCT was valid for the determination of vibrational quenching cross sections of BaRb⁺(v) + Rb at energies above 1 mK. ,, In 2019, Pérez-Ríos considered interaction of spin-polarized Rb atoms in terms of Strauss et al.’s empirical triplet a Σ _u potential and interaction of Ba⁺ – Rb according to a generalized Lennard-Jones potential that behaves asymptotically as a fourth-order van der Waals potential. QCT was again calculated with nuclear dynamics treated according to classical mechanics with initialization of trajectories and identification of final trajectories as individual quantum states treated according to the semiclassical quantization rule. The scaling of the vibrational quenching cross section was found to be consistent with the classical Langevin capture model scaling where the molecule was initialized in vibrational states v = 187–195 with angular momentum j = 0 for collision energies at and above 1 mK, with discrepancies for higher initial vibrational states. In 2021, Mohammadi et al. subsequently found that quasiclassical mechanics agrees with the classical Langevin value for BaRb⁺ (v)+ Rb for (2)¹ Σ⁺ and (1)³ Σ⁺ vibrational relaxation cross sections. Mohammadi et al. employed QCT to simulate the collision, as in Pérez-Ríos’ 2019 approach, using a pairwise additive ground-state potential with a Rb₂ interaction given by Strauss et al. and generalized Lennard-Jones potentials for both Ba⁺ – Rb and Ba – Rb⁺. The vibrational relaxation cross section from QCT was shown to be near the Langevin value for vibrational quantum numbers v = 5–16 for all collision energies studied, namely in the range of 1 to 100 mK. And, in 2024, Koots and Pérez-Ríos revisited the RbBa⁺ + Rb collision, now with the use of a Lennard-Jones potential to represent the triplet a Σ _u state of Rb₂ with QCT performed with their Python Quasi-Classical Atom-Molecule Scattering (PyQCAMS) package. These results confirmed quenching cross sections in agreement with the classical Langevin capture model at 20 and 100 mK for v = 187–195, with closest agreement for the lower vibrational quantum numbers. Over the years 2014 to 2021, Pérez-Ríos et al. have further employed a classical model based on a dynamical Langevin approach to determine classical ultracold threshold laws in a wide array of literature on three-body recombination ,,,, for which Dieterle et al. have experimentally confirmed the presence of Langevin capture dynamics for inelastic three-body collisions of individual ionic impurities in high-density rubidium Bose–Einstein condensates and Stevenson et al. have experimentally observed close agreement between NaCs three-body loss rates for varied field detunings and Rabi couplings across temperatures of 100 to 500 nK. ,,

2.1.2. Experimental Evidence

The aforementioned numerical evidence predicts quantum-to-classical crossover in the cold to ultracold regime for a broad range of atom–atom, atom–ion, molecule–molecule, and atom–molecule collisions. ,,,,,− ,− ,, Fortunately, in the past 15 years, experiment has supported these numerical predictions with flying colors. − ,,,− ,,,− A wide variety of experimental setups, including optical and/or radio-frequency traps ,,,,,, and merged supersonic beams, ,,, have reliably produced observations in keeping with classical and semiclassical predictions for disparate atomic and molecular processes including quenching, , charge exchange, ,, photofragmentation, , and atom–ion/molecule capture. ,,,

2.1.2.1. Atom–Atom and Atom–Ion Collisions

In 2024, Xing et al. found that total cross sections for Li­(2S _1/2) + Ba⁺ (5D _5/2) and Li­(2S _1/2) + Ba⁺ (5D _3/2) at 30 μK are consistent with scaled classical Langevin cross sections both in experiment and quantum numerics, as shown in Figure . This finding further supports the ability of classical mechanics to accurately describe many atom–ion collisions in the ultracold regime. To measure the cross sections experimentally, Xing et al. employed a combined all-in-one-spot ultracold atom apparatus with a segmented linear Paul trap. , ¹³⁸Ba⁺ atoms were cooled to approximately the Doppler temperature of ∼354 μK and prepared in their ⁵ D _3/2 or ⁵ D _5/2 electronic state and ⁶Li atoms were cooled to approximately quantum degeneracy and prepared in their |f = 1/2, m _f = −1/2⟩ state. The survival probability of Ba⁺ (5² D _3/2) or Ba⁺ (5² D _5/2) was recorded after various interaction durations, as well as the proportion of fine-structure quenching (FSQ), nonradiative charge exchange (NRCE), and nonradiative quenching (NRQ) events associated with distinct radial, spin–orbit, and rotational energy level transitions. Numerically, calculations were performed for LiBa⁺ potential energy curves computed via the CIPSI (configuration interaction by perturbation of a multiconfiguration wave function selected iteratively) package − with spin–orbit couplings computed via a quasidiabatic method. − Quantum cross sections were calculated in space-fixed frame according to two models: (i) a four-channel quantum scattering (FCQS) model of states in body-fixed frame with total electronic angular momentum Ω = 0⁺: 1¹Σ⁺ of Li­(2s) + Ba⁺ (6s), 2¹Σ⁺ of Li⁺ + Ba­(6s ^{2 1} S), and 3¹Σ⁺ and 1³Π of Li­(2s) + Ba⁺ (5d) (with coupling between atomic internal angular momentum and relative motional angular momentum neglected) and (ii) a multichannel quantum scattering (MCQS) model that additionally included Ω = 0^–, 1 for 1³Σ⁺ of Li­(2s) + Ba⁺ (6s), as well as Ω = 0^–, 1 for 2³Σ⁺, Ω^– = 0^–, 1, 2 for 1³Π, Ω = 1 for 1¹Π, Ω = 1, 2, 3 for 1³Δ, and Ω = 2 for 1¹Δ of Li­(2s) + Ba⁺ (5s). Results were thermalized according to a Maxwell–Boltzmann velocity distribution. For the FCQS model, the classical Langevin cross sections provided an upper bound for quantum cross sections. For MCQD, classical Langevin cross sections no longer serve as an upper bound for quantum cross sections locally due to the impact of shape resonances, but qualitatively provide an energy scaling of the quantum cross sections for FSQ, NRCE, and NRQ processes, often at energies well below 10 mK. Xing et al. found that experimental NRCE and NRQ cross sections at 30 μK, normalized to the computed classical Langevin rate, agree with both quantum calculations and the classical partial Langevin rate coefficient. Note quantum numerics suggest that elastic contributions are significant in the collision and are greater than the total Langevin rate coefficient, which suggests that the full capture assumption of the standard classical Langevin model does not hold.

In 2009, Grier et al. demonstrated agreement between experimental and classical Langevin rate coefficients also extends to multiple isotope combinations of resonant charge exchange for Yb⁺ + Yb at collision energies as low as 3 μeV (35 mK). Grier et al. employed a double-trap setup , in which neutral and singly charged atoms were trapped with a magneto-optical trap (MOT) and surface planar Paul trap, respectively, in a shared spatial volume. Isotope-selective ion fluorescence was used to measure signatures of charge-exchange collisions of ^αYb⁺ + ^βYb →^αYb + ^βYb⁺ via tracking of ^αYb population decay. A pair of charge-coupled device (CCD) cameras were used to image the local atomic density n(r), which informed computation of the average atomic density colocated with the ions ⟨n⟩ = ∫p(r)n(r) dr for normalized ion distribution p(r) and subsequently the elastic rate coefficient K via a linear fit of the average atomic density ⟨n⟩ as a function of the measured ion decay rate constant Γ = 1/τ. For comparison, the semiclassical Langevin cross section was determined for an ab initio polarizability for the atomic ground state. , The rate coefficients were found to agree closely with the semiclassical Langevin value for collisions examined (¹⁷²Yb⁺ + ¹⁷⁴Yb, ¹⁷²Yb⁺ + ¹⁷¹Yb, and ¹⁷⁴Yb⁺ + ¹⁷²Yb). The agreement extended from 3.1 μeV (36 mK) to 4 meV (46 K), namely over more than 3 orders of magnitude in energy. At 100 μeV (1.2 K) the semiclassical prediction was 5.8 · 10^–10 cm³ s^–1 with an experimental measurement of 6 · 10^–10 cm³ s^–1an absolute difference of just O(10^–11 cm³ s^–1).

And, in 2021, Ben-shlomi et al. identified classical Langevin scaling of quenching cross sections in ⁸⁸Sr⁺ – Rb collisions at collision energies in the ultracold and cold regime as well. Ben-shlomi et al. measured the quenching cross section for the atom–ion collision with fine energy resolution on the order of 100 μK via the passage of optical-lattice-trapped Rb atoms across a radio-frequency-trapped ⁸⁸Sr⁺ ion. Measured total quenching, electronic-excitation-exchange, and spin–orbit-change cross sections closely agreed with the Langevin cross section for all energies measured, which appeared in the domain of [0.2, 12] mK. In 2017, Saito et al. also experimentally measured classical Langevin scaling in cross sections for cold ⁶Li and ⁴⁰Ca⁺ collisions. Saito et al. experimentally measured charge-exchange collision cross sections for the atom–ion mixture by trapping and cooling the two collision partners in distinct areas of a vacuum chamber and transporting the optically trapped atoms with an optical tweezer to be mixed with the radio-frequency-trapped ions. The resulting products of the collision were probed via measurement of the ions in their S _1/2, P _1/2, D _3/2, and D _5/2 states. The charge-exchange collision cross sections for the ions in a S _1/2, P _1/2, and D _3/2 mixture; the D _3/2 state alone; and the D _5/2 state alone agreed with the classical Langevin cross section for all energies measured, namely on the order of 1 mK to 1 K.

The extremely high accuracy possible with classical and quasiclassical treatment of cold molecules is especially evident in Kondov, Majewska et al.’s 2018 Sr₂ photofragmentation experiments. , Kondov, Majewska et al. answered a decades-old quandary in the field of photodissociation − by experimentally observing and theoretically confirming a quantum-to-quasiclassical crossover at temperatures on the order of mK in photofragment angular distributions of diatomic strontium. , Above this crossover, at collision energies on the order of 0.1 to 1 mK, experiment, quantum, and semiclassical photofragmentation distributions were often nearly visually indistinguishable, as shown in Figure . In particular, there was near complete agreement between fully quantum, semiclassical WKB, and quasiclassical axial recoil limit photofragment distributions for the 0 _u (−2,1,0) and 1 _u (−2,1,0) states at just 48 mK, as is readily visible in Figure (right column center and bottom)a finding that evidences the ability of semiclassical and quasiclassical mechanics to accurately model not only collision cross sections, but photofragment distributions at cold temperatures for suitable molecular systems.

9.

Comparison of measured, quantum numerical, and semiclassical WKB numerical photofragment angular distributions of cold and ultracold strontium for initial 0 _u (left column) and 1 _u (right column top) states at energies 13 MHz (0.6 mK), 33 MHz (1.6 mK), 53 MHz (2.5 mK), 100 MHz (4.8 mK), and 300 MHz (14.4 mK) where each pair of rows corresponds to experiment (top) and quantum theory (bottom) with the exception of the final semiclassical WKB row. Strong agreement is also visible between between quantum theory, the semiclassical WKB approximation, and the quasiclassical axial recoil limit results at 1000 MHz (48 mK) for 0 _u (right column center) and 1 _u (right column bottom), and between quantum theory (solid lines) and the semiclassical WKB approximation (dashed line) for R (thick red line) and cos δ (thin blue line) above energies of approximately 1 mK (bottom center). Additional details in refs , . Reproduced figures with permission from I. Majewska, S. S. Kondov, C.-H. Lee, M. McDonald, B. H. McGuyer, R. Moszynski, and T. Zelevensky, 98, 4, 043404, 2018, Copyright 2018 by the American Physical Society, and from S. S. Kondov, C.-H. Lee, M. McDonald, B. H. McGuyer, I. Majewska, R. Moszynski, and T. Zelevensky, 121, 14, 143401, 2018, Copyright 2018 by the American Physical Society.

To identify this agreement, ⁸⁸Sr atoms were cooled and photoassociated to produce Sr₂ molecules with a temperature on the order of μK in a one-dimensional optical lattice trap, and photodissociation light pulses with a duration of 10 to 20 μs were employed, leading to a time-of-flight of 20 to 800 μs and an imaging pulse of 10 to 20 μs. The inverse Abel transform was used to convert the time-of-flight images to cross sections, which were subsequently averaged radially to produce the angular photofragment density. To identify the quantum-to-quasiclassical crossover, the photodissociation cross sections were simulated fully quantum mechanically according to Fermi’s golden rule on ab initio potentials, , semiclassically according to the WKB approximation, and semiclassically via an alternative model that treats molecular rotation classically. A quasiclassical model was also used to predict the angular distribution as a function of the initial molecular orientation probability density and the angular probability density of electric-dipole light absorption. ,, The quantum approach was found to accurately determine the cross sections at all energies considered (modulo potential energy surface corrections).

The quasiclassical model accurately yielded the angular distribution for 0 _u (v = −4, J _i = 1, M _i = 0) molecules with parallelly polarized photodissociation light (P = 0) at the axial recoil limitthe energy above all potential energy barriers at which the photodissociation speed greatly surpasses rotation speeds, such that molecular fragments exit on the molecular axis, here just 14 mK. Likewise, the energy evolution of the amplitude R of photofragments in the J _i – 1 angular momentum state and the phase difference δ obeyed the quasiclassical axial recoil limit R ≈ 1/2 above 5 mK and reached the semiclassical WKB (as well as semiclassical model) values at ≳ 1 mK, as depicted in Figure . Note quantum nuclear spin statistics and selection rules still can lead to discrepancies relative to quasiclassical results even at high energies, as Kondov, Majewska et al. observed, for example, for 1 _u (−1,1,0) molecules subject to a projection quantum number selection rule ΔM = ±1 for perpendicular polarization that disallows odd angular momenta J. ,

Significant literature also points to the ability of classical and semiclassical approaches to reproduce quantum mechanical and experimental results for cold and ultracold atom–atom and atom–ion collisions in the presence of electromagnetic fields. ,− Initial demonstrations in the late 1980s and 1990s ,− indicated agreement between (semi)­classical, quantum, and experimental analyses of atomic loss in optical traps with molecular states subject to laser-field dressing at collision energies on the order of 100 μK. − In the late 2000s and 2010s, classical and semiclassical methods to simulate cooling and trapping in radio frequency ion traps were found to be consistent with experiments. ,,− Notable among these advancements include Monte Carlo simulations based on classical time-evolution matrices by Devoe and classical trajectory simulations based on hard-sphere collisions and the polarization method of Cetina, Grier, and Vuletić by Meir et al. Devoe’s simulations closely reproduced experimental trapped ion lifetimes in noble buffer gases, including those of ¹³⁶Ba⁺ in argon (theoretical value, 45 s; experimental value, 50 to 100 s) and krypton or xenon (theoretical value, < 0.1 s; experimental value, <5 s). ,, Likewise, Meir et al.’s simulations closely reproduced experimental values for the dependence of the ion temperature in the range of approximately 0.5 to 3 mK on the number of Langevin collisions for an ⁸⁸Sr⁺ ion in a linear segmented Paul trap with ∼5 μK ⁸⁷Rb in a cross dipole trap. Furthermore, all of Meir et al.’s classically simulated data points for the collision-number dependence of the ion temperature were found to be within one-sigma confidence bounds of the experimental mean. Contemporaneously, Wright et al. further supported the ability of (semi)­classical mechanics to accurately model experimental loss rates of ultracold ions in traps with pulsed, frequency-chirped light close to resonance, ,− as their hybrid Landau–Zener excitation and Monte Carlo calculation based on classical atomic dynamics produced qualitative agreement with both experiment and fully quantum simulations. There are also studies in the literature that examine these systems from the perspective of semiclassical catastrophes and rainbows, , alternative classical models of cooling in electromagnetic traps, − and semiclassical models of quantum beats, which are interesting to watch as additional examples.

2.1.2.2. Molecule–Molecule Collisions

Just as predicted by quantum theory, Höveler et al. likewise experimentally observed classical capture rate coefficients on the order of 10 mK for molecular collisions of H₂ + HD and on the order of 1 K for molecular collisions of H₂ + D₂, H₂ + H₂, and HD⁺ + HD, as shown in Figures , , and . ,, This finding represents classical behavior well within and on the cusp of the cold regime in molecule-cation collisions of a family of diatomic hydrogen molecule isotopologues, as presaged by Dashevskaya et al. The collision was studied experimentally via a merged supersonic beam setup: ,, Höveler et al. prepared ground rotational state J = 0 molecules (GS) in two pulsed supersonic beams, excited one beam’s molecules to a low-field seeking Rydberg–Stark state (Rg) with a rovibronic ground state ion core of a given principal quantum number, and merged the beams with a relative mean velocity of v _rel = v _Rg – v _GS. The product ions were then measured with a time-of-flight mass spectrometer. The resulting relative total reaction cross section and rate coefficients for all collisions were compared to the classical Langevin capture model scaling, with H₂ + H₂, H₂ + D₂, and HD⁺ + HD collisions additionally compared against the theoretical rate coefficients of Dashevskaya et al. with appropriate mass scaling for isotopologues as required. ,,

10.

Close agreement between the classical Langevin law [exponent 0.5] and experiment [exponent – 0.505(14) from linear regression, black line] for the relative total reaction cross section in the H₂ + HD collision. Experimental data are shown for time-of-flight windows of 5 μs (blue points) and 6 μs (red points), with vertical and horizontal gray bars indicating standard deviation of single data points and extent of probed collision energies, respectively. Used with permission of the Royal Society of Chemistry from The H₂ + HD reaction at low collision energies: H₃/H₂D⁺ branching ratio and product-kinetic-energy distributions, K. Höveler, J. Deiglmayr, J. A. Agner, H. Schmutz, and F. Merkt, 23, 4 2021; permission conveyed through Copyright Clearance Center, Inc.

11.

Quantum-to-classical crossover apparent in the cold regime for (top) ${\mathrm{H}}_2^{+}+{\mathrm{D}}_2$ and (bottom) ${\mathrm{H}}_2^{+}+{\mathrm{H}}_2$ . Gray points indicate individual measurements and black points measurements averaged within collision-energy ranges delineated with gray vertical lines, where vertical bars indicate one standard deviation and dashed lines indicate theoretical predictions. A transition is evident below 1 K between the quantum threshold value [blue arrow] and the classical Langevin capture model limit [red solid horizontal line]. Figure used with permission from Höveler, Deiglmayr, and Merkt from ref , CC BY 2021 Taylor & Francis.

12.

Cold quantum-to-classical crossover below 1 K visible in experimental rate coefficients for (a–c) H₂ + H₂ collisions [with para H₂ (J = 0):ortho-H₂ (J = 1) ratios of (a) 25:75, (b) 59(1):41(1), and (c) 80(1):20(1)] and (d) HD⁺ + HD collisions with 4.25% H₂ and 4.25% D₂ by volume. Theoretically computed contributions of individual molecules/molecular states are shown in color ([a–c]: ortho-H₂ (J = 1) in red, para-H₂ (J = 0) in blue, [d]: HD (J = 0) in blue, HD (J = 1) in red, H₂ in green, D₂ in orange), with experimental data and classical Langevin values presented as in Figure . Figure used with permission from Katharina Höveler, Johannes Deiglmayr, Josef A. Agner, Raphaël Hahn, Valentina Zhelyazkova, and Frédéric Merkt, Physical Review A, 106, 5, 052806, 2022, Copyright 2022 by the American Physical Society.

These experimental results showed clear agreement with classical Langevin capture model predictions both in terms of reaction cross sections and rate coefficients. ,, The experimental relative total reaction cross section for H₂ + HD agreed with the classical Langevin capture law for all collision energies E _coll measured50 mK to 30 K, encompassing 3 orders of magnitudewith a linear regression slope −0.505(14) in agreement with Langevin capture E _coll with an absolute difference of O(10^–3) (i.e., a relative discrepancy of just 1%). The experimental rate coefficients for H₂ + H₂ and H₂ + D₂ agreed with the classical Langevin capture scaling at higher energies above 3 K. Moreover, analysis of the deviation from classical Langevin behavior in the quantum limit below 1 K supported the theoretical prediction that a lower reduced mass would be associated with a wider collision energy range associated with deviations from Langevin behavior. , Namely, quantum theory predicts a 3.0 enhancement factor in the threshold rate coefficient’s rise between H₂ and D₂ (due to a product of the relative proportion of ortho and para spin isomers in the natural molecules and the reduced mass ratio for D₂ and H₂, $\frac{9}{4}·\frac{4}{3}=3.0$ ), which was supported by the experimental finding of an enhancement factor of 2.8(3), a difference of 7%. Additionally, the experimental branching ratio of H₂ + D₂ into H₂D⁺ + D and HD₂ + H products at 10 K of η = 0.341(15) and 1 – η agreed with the quasiclassical prediction of 0.3 at 0.01 meV (1.2 K), an absolute prediction error of O(10^–2). Experimental rate coefficients for H₂ + H₂ and HD⁺ + HD were both consistent with reaching the classical Langevin capture model scaling by a collision energy on the order of 1 mK and consistent with Vogt and Wannier’s prediction over half a century prior: that ion-atom/molecule capture is enhanced by a factor of 2 at the quantum limit with respect to the Langevin rate. Finer collision-energy resolution remains a need to verify the precise enhancement factor.

2.1.2.3. Atom–Background Gas Collisions

In the same vein, for glancing and diffractive collisions of optically and magnetically trapped atoms and background gases, a wide array of authors have found close agreement between classical and semiclassical formulas ,,− and experimental data. , In 1988, Bjorkholm built on the classical small-angle scattering formula ,− to model trap loss due to elastic collisions of cold atoms with ambient background gas molecules as occurring due to the transfer of sufficient energy to expel the atom from the trap (namely, where the scattering angle exceeds the associated threshold). ,, This classical ejection rate yielded lifetimes that closely matched experimental values for sodium in varied conditions: 500 mK magnetic-molasses optical traps at pressure p (theoretical value, 2.2 · 10^–8 s/p; experimental value, 2.0 · 10^–8 s/p), 17 mK magnetic traps at pressure 10^–8 Torr (theoretical value, 1.1 s; experimental value 0.83 s), and ∼5 mK optical traps at pressure 4 · 10^–9 Torr (theoretical value, 2.2 s; experimental ∼ 1 s). Likewise, Arpornthip, Sackett, and Hughes’ 2012 classical loss rate for atoms in 1 K traps colliding with 300 K ambient gases closely agreed with experiment for sodium in hydrogen gas (theoretical value, 5.3 · 10⁷ Torr^–1 s^–1; experimental value, 5 · 10⁷ Torr^–1 s^–1), as well as for rubidium in argon gas (theoretical value, 2.3 · 10⁷ Torr^–1 s^–1; experimental values 2.2 · 10⁷ Torr^–1 s^–1 and 1.6 · 10⁷ Torr^–1 s^–1). , Arpornthip, Sackett, and Hughes’ classical formula additionally concurred with the experimentally observed behavior of reduced dependence of the loss rate on the trap depth in the range [0.5, 2] K , with deviations observed for shallower traps where the classical approximation fails. ,,,

In the 2010s, Booth, Shen, Krems, and Madison similarly developed a semiclassical approach to trapped atom–background gas collisions that agrees with both quantum calculations and ultracold experiments to high accuracy. Booth, Shen, Krems, and Madison employed the minimum angle leading to trapped atom ejection and the total collision rate expression for a Maxwell–Boltzmann distribution to produce a semiclassical loss rate Γ_loss = n⟨σ_tot v⟩(1 – p _QDU) as a function of the gas density n, total cross section σ_tot, collision velocity v, and a novel probability of atomic retention in the trap p _QDU. This probability p _QDU was found to constitute a law of quantum diffraction universality $p_{\mathrm{QDU}}\equiv \sum _{j=1}^{\infty }{\beta }_j{(U/U_d)}^j$ in terms of the trap depth U, in which the coefficients β _j and characteristic quantum diffractive energy U _d are entirely independent of the van der Waals coefficient, mass, and dipole (ultimately leading to a universal formula for the gas pressure P = nk _B T = Γ_loss (U)k _B T/⟨σ_loss (U)v⟩ that has led to a search extending into the 2020s to establish the use of cold trapped atoms as pressure sensors ,− ,− ).

Booth, Shen, Krems, and Madison’s semiclassically predicted pressure agreed with experiment for magnetically trapped ⁸⁷Rb atoms at <0.5 mK with helium, xenon, argon, molecular hydrogen, nitrogen, and carbon dioxide. In particular, in the case of nitrogen, the associated pressure measurement reached as high as 0.5%-level agreement with the value for an ionization gauge calibrated by the National Institute for Standards and Technology (NIST). Similarly, in 2023, Kłos and Tiesinga further found agreement between quantum scattering results and the semiclassical small-angle approximation to the thermalized total elastic rate coefficient to within 10% for ultracold ⁷Li and ⁸⁷Rb with ⁴He, ²⁰Ne, ⁴⁰Ar, ⁸⁴Kr, ¹³²Xe, and ¹⁴N₂; ,− and, in 2024, Booth and Madison identified agreement between the quantum scattering results of Medvedev et al. and Kłos and Tiesinga and a semiclassical formula based on the Jeffreys–Born approximation for accurate total rate coefficients for collisions of cold ⁸⁷Rb and ⁷Li with higher-mass background gases, in the case of trapped ⁷Li, with an error of less than 2% for krypton or xenon background gas and with an error less than the uncertainty range for argon or nitrogen background gas. These results for diffractive and glancing collisions thus support the use of semiclassical and classical methods to produce near-exact results in the cold and ultracold regimes for a wide array of atomic and molecular systems.

2.1.2.4. Atom–Molecule Collisions

We see the same high level of correspondence between (semi)­classical methods and experiment in cold and ultracold atom–molecule collisions. With less than 1% error from classical predictions, Shagam et al. found in merged supersonic beam experiments that the rate constants for reaction of para-H₂ (j = 0) and HD with He­(2³ P ₂) obey the classical Langevin law to temperatures as low as on the order of 0.1 K, as shown in Figure further evidence of classical behavior in cold collisions of atoms with hydrogen molecule isotopologues. The experiment prepared supersonic beams of H₂ or HD and He, excited He to the desired collision partner state, and employed a magnetic quadrupole guide to merge the resulting beams. The resulting beams were collided in a time-of-flight mass spectrometer such that the reaction rate could be found via detection of the atom at a microchannel plate. − For comparison, the scaling of the respective classical Langevin/Gorin scaling law was determined according to the order of the long-range van der Waals interaction, with the rate itself determined according to an estimated van der Waals coefficient. For collision of para-H₂ with He­(2³ P ₂), which features an asymptotic sixth-order van der Waals interaction, the sixth-order van der Waals coefficient was determined according to the H₂ dipole polarizability of Bishop and Pipin and the sum-over-states expression for He­(2³ P ₂) polarizability, dipole moments and transitions were calculated according to multireference configuration interaction calculations with single and double excitations (MRCISD), and potential energy surfaces were produced with the MOLPRO and DALTON packages. For collisions of He­(2³ P ₂) with para-H₂, the reaction rate constant scaled according to classical Langevin (Gorin) scaling to energies as low as 0.8 K, namely, in the unitary “black hole” limit in which reactants capable of surpassing the centrifugal energy barrier formed products with (near-)­unit probability and the rate depended solely on the long-range interaction. The energy dependence of the reaction rate constant for collisions of He­(2³ P ₂) with the HD isotopologue was also found to be consistent with the same classical Langevin scaling. Shagam et al. noted that the scaling law is independent of collision partner for a given specific van der Waals interaction, and suggested that the conformity of experimental reaction rates with universal Langevin scaling laws allows such laws to be used to probe long-range forces in cold collisions, including those that pertain to astrochemistry.

2.2. Rationales for the Accuracy of Classical and Semiclassical Approaches in the Cold and Ultracold Regime

Recent theoretical and experimental work (refs , , , , , , , , and − ) suggests a variety of rationales for why classical behavior of near-threshold physical and chemical systems is so ubiquitous beyond the classical limit of ℏ→0, high energy, and large quantum numbers ,,− ,,,, and how to predict what new ultracold molecular systems are expected to be accurately simulable with computationally efficient classical, semiclassical, and quantum-classical methods in the future.

2.2.1. Approach of Gribakin and Flambaum

Since 2020, a plurality of studies (see refs , , , , , − , − , − , − , − , − ) relate semiclassical behavior in cold and ultracold atomic and molecular collisions to the 1993 work of Gribakin and Flambaum, who discovered that semiclassical mechanics is often sufficient to accurately determine cross sections even for low collision energies typically associated with quantum mechanics. ,

Initially, Gribakin and Flambaum argued similarly that the semiclassical WKB approximation is expected to be accurate for scattering at low collision energies as long as there exist deep potential wells that permit the wave function to have a short oscillation period. This argument permits direct determination of the scattering length from the connection of the semiclassical wave function in the interior region to the quantum mechanical wave function in the exterior region at the region boundary. Specifically, Gribakin and Flambaum first assumed the semiclassical WKB solution holds in the interior beyond the classical turning point

$$\chi (R)=\frac{C}{\sqrt{p}}\cos (\frac{1}{\hslash }{\int }_{R_0}^{R}p\mathrm{d}R-\frac{\pi }{4}),R>R_0$$ 3

and a quantum superposition state of Bessel and Neumann functions holds in the exterior asymptotic region. Matching the semiclassical WKB solution to the approximate analytic solution for an n ^th order asymptotic van der Waals potential U(R) ≃ −α/R ⁿ at a radius R = R ^* at which the semiclassical interior and quantum exterior solutions agree yielded the near-threshold scattering length in terms of the mean scattering length a̅

$$a=\mathrm{a̅}[1-\tan \left(\frac{\pi }{n-2}\right)\tan (\mathrm{\Phi }-\frac{\pi }{2(n-2)})]$$ 4

$$\begin{array}{ll}\mathrm{a̅}& =\cos \left(\frac{\pi }{n-2}\right){\left(\frac{\gamma }{(n-2)}\right)}^{2/(n-2)}\\ & \times \mathrm{\Gamma }\left(\frac{n-3}{n-2}\right)/\mathrm{\Gamma }\left(\frac{n-1}{n-2}\right)\end{array}$$ 5

$$\mathrm{\Phi }=\frac{1}{\hslash }{\int }_{R_0}^{\infty }\sqrt{2M[-U(R)]}\mathrm{d}R$$ 6

for the zero-energy semiclassical phase between the classical turning point and infinity Φ and the derived constant $\gamma =\sqrt{2M\alpha }/\hslash$ . As predicted by the rationale that semiclassics holds for deep potential wells, the hybrid semiclassical-quantum method gave a scattering length that agrees closely with the quantum result (namely, the extrapolation of the quantum result to zero Kelvin), as well as experimental data, for scattering from the deep Cs–Cs³Σ _u potential (see Table )­

$$U(R)=\frac{1}{2}B{R}^{\alpha }{\mathrm{e}}^{-\beta R}-(\frac{C_6}{R^6}+\frac{C_8}{R^8}+\frac{C_{10}}{R^{10}})f_c(R)$$ 7

$$f_c(R)=\mathrm{\Theta }(R-R_c)+\mathrm{\Theta }(R_c-R){\mathrm{e}}^{-{(R_c/R-1)}^2}$$ 8

$$\mathrm{\Theta }(x)=\{\begin{array}{ll}1& x>0\\ 0& x<0\end{array}$$ 9

where the potential is given by the sum of an exchange repulsion term (parametrized by B, α, and β) and a series of van der Waals terms (parametrized by C ₆, C ₈, and C ₁₀ and modulated by a cutoff potential f _c (R) that removes the van der Waals terms’ R = 0 divergence via imposition of a cutoff radius R _c ). This scattering length in turn yielded a high-accuracy s-wave phase shift δ₀ = −ak at wavenumber k, as well as the near-threshold scattering cross section σ = 4πa ². Their subsequent development of a hybrid semiclassical-quantum formula for the effective range r _e in terms of the aforementioned γ and ν = 1/(n – 2) along the same lines continued to support this initial deep potential well argument. Based on the same argument that the wave function can be determined by matching the semiclassical wave function in the interior region and the quantum wave function in the exterior region at the region boundary, Flambaum and Gribakin found that the semiclassical equation for the effective range r _e

$$r_e=F_n-\frac{G_n}{a}+\frac{H_n}{a^2}$$ 10

$$F_n=\frac{2}{3}\frac{\pi }{\sin \nu \pi }{(\gamma \nu )}^{2\nu }\frac{\mathrm{\Gamma }(\nu )\mathrm{\Gamma }(4\nu )}{{[\mathrm{\Gamma }(2\nu )]}^2\mathrm{\Gamma }(3\nu )}$$ 11

$$G_n=\frac{4}{3}\frac{\pi }{\sin \nu \pi }{(\gamma \nu )}^{4\nu }\frac{\mathrm{\Gamma }(1-2\nu )\mathrm{\Gamma }(4\nu )}{\nu \mathrm{\Gamma }(\nu )\mathrm{\Gamma }(2\nu )\mathrm{\Gamma }(3\nu )}$$ 12

$$H_n=\frac{2}{3}\frac{\pi }{\sin \nu \pi }{(\gamma \nu )}^{6\nu }\frac{\mathrm{\Gamma }(1-3\nu )\mathrm{\Gamma }(1-\nu )\mathrm{\Gamma }(4\nu )}{{\nu }^2{[\mathrm{\Gamma }(\nu )]}^2{[\mathrm{\Gamma }(2\nu )]}^2}$$ 13

produced values of the effective range r _e for the deep potentials of Li₂, Na₂, and Cs₂ that agreed with quantum results with less than 1% error, again based on zero-energy semiclassical and quantum solutions. Note the special case result for n = 6 was simultaneously developed by Gao in the context of quantum defect theory. This effective range in turn gave high-accuracy s-wave phase shifts according to the definition of the standard effective range expansion

$$k\cot {\delta }_0\simeq \frac{1}{a}+\frac{1}{2}{r}_e{k}^2$$ 14

5. Gribakin and Flambaum’s Semiclassical Scattering Length a _class (eq ) Closely Agrees with the Fully Quantum Mechanical Scattering Length a _quant with Low Relative Error for Various Parametrizations of the Cs₂ ³Σ _u Potential (eq ) .

R c [au]	Rmin [au]	Umin [10–3 au]	aclass [au]	aquant [au]	Relative Error
23.215	11.93	–1.273	376	352.5	6.7%
23.190	11.91	–1.286	140	144.2	2.9%
23.165	11.88	–1.300	65	68.0	4.4%
23.140	11.86	–1.313	–69	–67.7	1.9%
23.115	11.83	–1.327	467	485.3	3.8%

^aResults are shown for parameters B = 0.0016, α = 5.53, β = 1.072, C ₆ = 6.5, C ₈ = 1.1 · 10⁶, and C ₁₀ = 1.7 · 10⁸ from ref , with stated cutoff radii R _c resulting in shifted potential minima (R _min, U _min). The quantum result follows from extrapolation of the s-wave phase shift at k = 0.005 au (12 μK) to k = 0 au (zero Kelvin). Reproduced table with permission from G. F. Gribakin and V. V. Flambaum, Physical Review A, 48, 1, 546, 1993. Copyright 1993 by the American Physical Society.

The combination of these successes at first led Gribakin and Flambaum to first formally state that the scattering phase shift for deep potentials with asymptotic behavior U (R) ≃ −α/R ⁿ depends only on the derived constant γ and the semiclassical phase Φ where the potential well depth is much greater than the scattering energy. This implies the scattering phase shift in such cases is independent of energy, which they connect to the fact that deep in the potential well the potential may be substituted with a boundary condition that is independent of the energy.

However, what most closely relates to recent measurement of semiclassical behavior in cold and ultracold collisions is that Gribakin and Flambaum subsequently identified that, beyond their deep well hypothesis, their semiclassical method also succeeds for determination of the scattering length for a shallow H₂ ³Σ _u potential. Gribakin and Flambaum rationalized the close agreement between semiclassical and quantum results for this shallow potential in terms of their derived formula and noted that the scattering length a for the system is significantly smaller than the average scattering length a̅, which ensures the scattering length is determined nearly entirely by the semiclassical phase Φ between the classical turning point and infinite radius. This finding points to a potentially much broader range of applicability of semiclassical mechanics near threshold in modern studies than originally theorized in which the semiclassical approximation can provide accurate results for shallow potentials, including potentials that support no wave function oscillations.

2.2.2. WKB Criterion

The WKB approximation has also been used to explain quantum-to-classical crossovers in the cold and ultracold regimes. These rationales, introduced by Julienne and Mies in 1989, Côté et al. in 1996, and Mody et al. in 2001, continue to be actively cited nearly three decades on (see refs , , , , − , − , , , − , , , , , , , , , − ).

Specifically, Julienne and Mies proposed that one may identify a broad range of energies at which semiclassical methods are accurate near threshold via careful consideration of the energies for which WKB holds at all interatomic distances. Building on prior work, − Julienne and Mies began with considering the deep connection between WKB and multichannel quantum defect theory (MQDT), which revealed that MQDT reference wave functions approach WKB wave functions at energies sufficiently beyond threshold, as follows: In MQDT, the fully quantum MQDT reference functions in the interior potential for the analytic matrix Y(E) at energy E, wavenumber k, and radius R assume the WKB-like form

$$f={\sqrt{K(E,R)}}^{-1}\sin (\beta (E,R))$$ 15

$$g={\sqrt{K(E,R)}}^{-1}\cos (\beta (E,R))$$ 16

$$K(E,R)=\sqrt{\frac{2\mu (E-U^0(R))}{{\hslash }^2}}$$ 17

$$\gg k$$ 18

for an uncoupled channel reference potential U ⁰ and channel state β; and quantum mechanically the reference wave functions for the scattering matrix S(E) assume the form

$$\underset{R\to \infty }{\lim }s(E,R)\sim k^{-1/2}\sin (kR-\pi l/2+\xi )$$ 19

$$\underset{R\to \infty }{\lim }c(E,R)\sim k^{-1/2}\cos (kR-\pi l/2+\xi )$$ 20

for relative angular momentum of the collision partners l and scattering phase shift ξ. Julienne and Mies noted that for great enough collision kinetic energy ε, the MQDT reference functions f and g become equal to s and c, respectively. Moving past the deep-well theorem, Julienne and Mies clarified that the criterion for this agreement is that the WKB approximation holds for all radii: namely, that the local de Broglie wavelength Λ­(E,R) varies slowly as

$${\mathrm{\Lambda }}^{\prime }=\frac{\mathrm{d}\mathrm{\Lambda }(E,R)}{\mathrm{d}R}\ll 1$$ 21

To identify the range of energies at which this approach applies, Julienne and Mies introduced a graphical approach, thereafter adapted by others, , which plots the criterion Λ′ as a function of the radii R for varying collision energies ε/k _B (see Figure ). The energy at which semiclassical mechanics breaks down and a quantum-to-classical crossover occurs is then approximated as the energy at which the criterion rises above a benchmark value at any radius. Below this energy, Julienne and Mies attributed the breakdown of semiclassical treatment and appearance of quantum effects as arising from the loss of a smooth WKB connection between the WKB-like solution in the inner region and the asymptotic wave function. Above this energy, the WKB approximation is expected to closely approximate the quantum wave function at all radii and to thereby yield an accurate depiction of the associated scattering processeven in cases where the energy satisfying the WKB criterion falls in the cold or ultracold regime.

14.

WKB criterion for the (a) $\mathrm{He}({}^{3}S)+\mathrm{He}({}^{3}S)$ collision and (b) neon impinging on a semi-infinite SiN slab indicating the accuracy of semiclassical methods to energies as low as approximately 1 K and 200 nK, respectively. Adapted with permission from ref . Copyright Optical Society of America and from A. Mody, M. Haggerty, J. M. Doyle, and E. J. Heller, Physical Review B, 64, 8, 085418, 2001. Copyright 2001 by the American Physical Society.

Note Julienne and Mies’ argument suggests that semiclassical mechanics is expected to be accurate well within the ultracold regime for a number of atomic collisions for ground and metastable state species, as summarized in Table . To approximate the characteristic quantum threshold energy ε _Q below which semiclassical treatment breaks down within a given range of radii, Julienne and Mies identified the maximum WKB criterion for an asymptotic potential V ⁰ (R) → −C _n /R ⁿ for n ≥ 3

$${\mathrm{\Lambda }}_{\max }^{\prime }(\epsilon ,R_Q)=\frac{(n+1)}{3}{\left[\frac{(n-2){\hslash }^2}{6n\mu \epsilon }\right]}^{1/2}{\left(\frac{n+1}{n-2}\frac{2\epsilon }{C_n}\right)}^{1/n}$$ 22

at radius

$$R_Q={\left(\frac{n-2}{2n+2}\frac{C_n}{\epsilon }\right)}^{1/n}$$ 23

which yields the energy at which the WKB criterion begins to rise above the arbitrary maximum WKB criterion Λ_max = 1/2

$$\epsilon =\frac{{\hslash }^2}{\mu }{\left[{\left(\frac{n+1}{3}\right)}^{2n}{\left(\frac{n-2}{6n}\right)}^n{\left(\frac{2n+2}{n-2}\frac{{\hslash }^2}{\mu C_n}\right)}^2\right]}^{1/(n-2)}$$ 24

at effective temperature T _Q = ε _Q /k _B. Although strongly dependent on the force power n, reduced mass μ, and van der Waals constant C _n , this quantum threshold energy formula indicates semiclassical treatment of rate coefficients can remain accurate at ultracold temperatures of T > 300 μK for ground-state Cs collisions or T > 1 nK for $\mathrm{Na}({{}^{2}P}_{3/2})$ and ground-state Na collisions, as detailed in Table . This finding confirmed their prior calculation showing semiclassical and quantal agreement in the computed spectrum for $\mathrm{Na}({{}^{2}P}_{3/2})+\mathrm{Na}$ at 1 mK, work further expanded into the 2010s. − It is important to note that ε _Q is considered to be the upper bound of the quantum-to-classical crossover, as the crossover varies in response to the energy of the highest-energy bound vibrational state, with key considerations with reference to inelastic scattering collisions, as discussed by Siska in 2001. Siska showed that these caveats in fact allow the quantum-to-classical crossover to appear at orders of magnitude lower energy than even ε _Q for Ne⁺ + He. 15 years later, DeMille, Glenn, and Petricka would go on to show that this analysis predicts semiclassical behavior of SrO collisions in standing-wave electromagnetic-field microwave traps at temperatures above 200 fKor, where the upper bound of s-wave collisions is instead held as a benchmark, 30 fK.

6. Characteristic Quantum Threshold Temperatures T _Q above which Semiclassical Methods Hold, as Derived by Julienne and Mies for Collisions of Species in Specified States Subject to van der Waals Potentials of Force Power n .

Species	State	n	T Q
H + H	${{}^1\mathrm{\Sigma }}_g^{+}$	6	20 K
$\mathrm{He}({}^{3}S)+\mathrm{He}({}^{3}S)$	${{}^1\mathrm{\Sigma }}_g^{+}$	6	100 mK
Cs + Cs	${{}^1\mathrm{\Sigma }}_g^{+}$	6	300 μK
Na + Na	${{}^1\mathrm{\Sigma }}_g^{+}$	6	10 mK
$\mathrm{Na}({{}^{2}P}_{3/2})+\mathrm{Na}({{}^{2}P}_{3/2})$	1 u	5	740 μK
$\mathrm{Na}({{}^{2}P}_{3/2})+\mathrm{Na}$	0 u	3	1 nK

^aNote T _Q occurs in the ultracold regime for collisions of Cs + Cs, $\mathrm{Na}({{}^{2}P}_{3/2})+\mathrm{Na}({{}^{2}P}_{3/2})$ , and $\mathrm{Na}({{}^{2}P}_{3/2})+\mathrm{Na}$ and in the cold regime for $\mathrm{He}({}^{3}S)+\mathrm{He}({}^{3}S)$ and Na + Na. Reproduced with permission from ref . Copyright Optical Society of America.

Subsequent work on cold–atom collisions and atom–surface sticking , tied Julienne and Mies’ wide-ranging agreement between semiclassical and quantal results in the ultracold regime to a physical origin: the loss of quantum reflection. ,,,− These works considered that the WKB approximation holds to first approximation where the WKB criterion is satisfied, stated in terms of the de Broglie wavelength λ­(x), momentum $p(x)=\sqrt{2m(E-V(x))}$ , or potential V (x) as

$$1\gg \frac{1}{2\pi }\left|\frac{\mathrm{d}\lambda (x)}{\mathrm{d}x}\right|$$ 25

$$1\gg \left|\frac{\hslash }{p(x)}\frac{\mathrm{d}p(x)}{\mathrm{d}x}\right|$$ 26

$$1\gg \frac{m\hslash }{{(p(x))}^3}\left|\frac{\mathrm{d}V(x)}{\mathrm{d}x}\right|$$ 27

for mass m and energy E. What distinguished their rationale is an explicit link between the failure of WKB and quantum reflection, which as Mody et al. stated in 2001 concerns the quantum effect “that quantum mechanically the amplitude is reflected back without penetrating the interaction region, analogous to the elementary case of reflection from the edge of a step-down potential in one dimension while attempting to go over the edge.” An illustrative example of such quantum reflection from 2018 , is given in Figure for context, specifically scattering from the attractive V(q) = 2tanh­[q/5]-2 step potential. In the quantum regime, semiclassical and quantum amplitudes differ, with the wave function reflected backward to have higher amplitude at right due to quantum reflection. As the collision energy increases, the quantum wave function passes through increasingly, until semiclassical and quantum amplitudes agree for energies at which the WKB criterion is satisfied at all positions such that semiclassical mechanics accurately describes the system. (As an aside, it bears mentioning that this quantum interference effect has an acoustic analog: for example, a trumpet bell is shaped to cause wavelength-dependent sound wave reflection ever further toward the end of the bell with decreasing wavelength, thus providing a much needed realignment of the harmonic structure of the instrument.)

15.

Illustration of quantum reflection in the absence of a barrier at increasing collision energies (in arbitrary units) in the V(q) = 2tanh­[q/5]-2 step potential. At the lowest energy (top left), the quantum wave function (thin black line) is reflected from the step potential (thick black line), leading to greater amplitude at right than the semiclassical WKB result (dashed line). As the energy increases (left to right, top to bottom), the quantum amplitude approaches the semiclassical amplitude until quantum-classical correspondence is reached. Used with permission of Princeton University Press from The Semiclassical Way to Dynamics and Spectroscopy, E. J. Heller, 2018; permission conveyed through Copyright Clearance Center, Inc.

Côté et al. further recognized in 1996 that near-threshold quantum reflection is intimately tied to near-threshold quantum suppression, a topic that continues to be of active interest in ultracold scattering. ,,,,− ,,, Côté et al. showed that near-threshold reflection suppresses the exterior amplitude of the quantum scattering wave function relative to semiclassical and classical predictions. This led the authors to both demonstrate that the WKB criterion (which they term the WKB error) acts as a harbinger of quantum suppression and to derive a quantum suppression factor that corrects the WKB approximation to the quantum cross section, which allows semiclassical mechanics to be used to accurately determine quantities such as dipole matrix elements yet closer to threshold. This method enabled accurate determination of dipole matrix elements for singlet and triplet ⁷Li₂ transitions deep in the ultracold regime, as shown in Figure , which further supports the ability of semiclassical mechanics to accurately simulate aspects of s-wave atom–atom scattering systems close to threshold. Note the corrected results remain accurate to ultracold energies as low as O(10 au) [O(1 μK)].

16.

⁷Li₂ dipole matrix elements |D _v (E)|² for the ν = 68 (a) singlet and (b) triplet transitions computed exactly (solid line), semiclassically (dotted line), and semiclassically with a correction factor (dashed line), which indicate strong agreement between the quantum result and the hybrid quantum-semiclassical result deep in the ultracold regime. Approximate agreement exists between exact and corrected semiclassical results at energies as low as O(10^–11 au) [O(1 μK)]. Reproduced figure with permission from R. Côté, E. J. Heller, and A. Dalgarno, Physical Review A, 53, 1, 234, 1996. Copyright 1996 by the American Physical Society.

2.2.3. Beyond the WKB Criterion

Modern semiclassical and quantum theory has increasingly brought into question whether the WKB criterion alone is sufficient to identify the onset of quantum behaviors and the breakdown of semiclassical approaches. ,,, In 2004 to 2017, Friedrich and Trost recognized that not only the WKB criterion, but also the quantality ,, or badlands function

$$Q(x)={\hslash }^2(\frac{3}{4}\frac{{(p^{\prime }(r))}^2}{{(p(r))}^4}-\frac{p^{″}(r)}{2{(p(r))}^3})$$ 28

$$|Q(x)|\ll 1$$ 29

must also be small to ensure the WKB solution to the Schrödinger equation simultaneously fulfills both the Schrödinger equation

$$0=u^{″}(r)+\frac{{(p(r))}^2}{{\hslash }^2}u(r)$$ 30

and the equation for the second derivative of the WKB wave function

$$\begin{array}{ll}0& =u_{\mathrm{WKB}}^{″}(r)+\frac{{(p(r))}^2}{{\hslash }^2}{u}_{\mathrm{WKB}}(r)\\ & +(\frac{p^{″}(r)}{2p(r)}-\frac{3}{4}\frac{{(p^{\prime }(r))}^2}{{(p(r))}^2})u_{\mathrm{WKB}}(r)\end{array}$$ 31

In 2021 and 2023, this badlands function was also been capitalized on in ultracold chemistry in the form of a WKB potential to reduce anomalous reflection in complex absorbing potential simulations and to investigate bound states in the continuum, reflectionless scattering modes, and parity-time reversal $(\mathcal{PT})$ symmetry with Bose–Einstein condensates.

More recently, both the WKB criterion and the quantality have become the source of controversy as an indication of the “source” of quantum reflection. , In 2017, Miret-Artés, Petersen, and Pollak indicated that the nonlocality of quantum scattering means that the region of large quantality (the “badlands region”) is not solely responsible for quantum reflection and that the full potential must be considered. As evidence, Mirét-Artes and Pollak demonstrated that numerically exact close-coupling calculations of quantum reflection probabilities and diffraction patterns agree with experimental results for helium scattering from a microstructured grating only when the full potential region is considered. They further recall that the potential V(x) = −sech² (x) features no quantum reflection at any energy, − despite the presence of a badlands region at low energies. Likewise, Petersen, Pollak, and Miret-Artés performed time-dependent calculations that demonstrate that at the lowest energies a quantum wavepacket impinging on a Morse potential reaches its maximal density at a position orders of magnitude greater than the maximum magnitude of the badlands function (see Figure ). For this reason, the authors argue that quantum reflection is better considered as a result of not local high quantality but reduced density caused by nonlocal destructive interference. It is important to note that this caveat on the quantality, which is consistent with prior theorems concerning the applicability of semiclassical methods close to threshold, serves as a reminder of the need for care in WKB criterion and quantality analyses, in particular to recognize that destructive interference, not the magnitude of the quantality, indicates where quantum reflection occursan essential ingredient in the quest to accurately identify cases in which ultracold molecular collisions are amenable to semiclassical study.

17.

Quantum (top row) and classical Wigner (bottom row) propagation (color heat map) of a coherent state initialized with energy E _init = 5 · 10^–9 in Morse potential V(z) = e^–2z – 2e^–z over time t indicates that a quantum wavepacket features negligible magnitude relative to its classical counterpart at positions up to ∼1000 times greater than the location of the maximum badlands function magnitude z _BF ≈ 20.7 (vertical dotted blue line). Right column images magnify short-time dynamics. Reproduced figure with permission from J. Petersen, E. Pollak, and S. Miret-Artés, Physical Review A, 97, 4, 042102, 2018. Copyright 2018 by the American Physical Society.

In 2018, ref further emphasized that quantum results in the ultracold regime may in fact agree with not only semiclassical but also classical mechanics for aspects of systems near threshold in what was termed “ultracold inferno theory” due to the reliance of the semiclassical propagator on classical quantities. The authors considered that the semiclassical Van Vleck–Morette–Gutzwiller propagator −

$$G_{\mathrm{sc}}(q,q^{\prime },t)=\frac{1}{\sqrt{2\pi \mathrm{i}\hslash }}\sum _k{\left|\frac{{\partial }^2{S}_k(q,q^{\prime },t)}{\partial q\partial q^{\prime }}\right|}^{1/2}{\mathrm{e}}^{\mathrm{i}/\hslash S_k(q,q^{\prime },t)-\mathrm{i}\pi {\nu }_k/2}$$ 32

assumes the general form of the semiclassical amplitude

$$A\propto \sum _j\sqrt{\mathrm{classical\; probability\; density}}{\mathrm{e}}^{\mathrm{i}/\hslash ·{\mathrm{classical\; action}}_j}$$ 33

where S _k is the classical action and ν _k is the Maslov index of classical path z and the intersection of finite-width position-space regions q and q′ (t) where the sum is taken over each classical path j. In the absence of quantum reflection, the semiclassical amplitude agrees with the quantum amplitude, as argued in previous rationales. What this form emphasizes is that in the additional absence of quantum interference, semiclassical and classical results also agree, and there exists a full quantum-classical correspondence.

To provide an example of the impact of quantum interference detailed in ref , this review considers two fundamental nonrelativistic scattering systems: Mott and Rutherford scattering (see Figure ). Recall that in three-dimensional nonrelativistic Mott scattering, an α particle and a helium nucleus approach each other with equal and opposite velocities. , Upon collision, the α particle deflects by an angle ∠ACD = θ. Although only this path brings the α particle to point D, an additional contribution to the differential scattering cross section must be considered: the deflection of the indistinguishable helium nucleus by an angle ∠BCD = π – θ. Since the two scattered bodies are indistinguishable, a detector situated at point D cannot distinguish between the two paths ACD and BCD, such that the paths interfere and there exists no quantum-classical correspondence. In contrast, in three-dimensional Rutherford scattering, , an α particle of a given initial momentum and impact factor deflects off a gold atom to reach a specific angle with a given final momentum in a fixed amount of time. Here trajectories that lead the gold atom to be deflected into the same final angle do not contribute to the scattering cross section as the two colliding bodies are distinguishable. Since the particles are distinguishable, the system is free of quantum interference and quantum-classical correspondence exists in the absence of quantum reflection. In fact, classical and quantum results for the cross section agree for Rutherford scattering at all energiesa special case of quantum-classical correspondence at threshold related to atomic fragmentation with possible future ramifications for ultracold atomic systems.

18.

Schematic of classical pathways contributing to the semiclassical amplitude and possible quantum-classical correspondence in two fundamental scattering systems: (a) nonrelativisitic Mott scattering, in which an α particle A scatters off a Helium nucleus B toward detectors D and E and (b) Rutherford scattering, , in which an α particle (solid lines) scatters off a gold atom (solid point, example trajectory of a gold atom scattering off an α particle shown with a dashed line). Adapted from J. M. Rost and E. J. Heller, Semiclassical scattering of charged particles, Journal of Physics B, 27, 7, 1387–1395. Copyright IOP Publishing. Reproduced with permission. All rights reserved.

2.3. Semiclassical Scaling Laws and Holistic Quantum Models

In the midst of this classical renaissance; new developments are also being made to use classical, semiclassical, and quasiclassical techniques to accurately simulate a wider range of chemical systems. A wide variety of authors have developed novel semiclassical scaling laws to describe the post-threshold regime in a wider range of atomic and molecular systems. ,,,,− ,, Quéméner, Gao, Bohn et al. have established holistic quantum-to-semiclassical models that closely model atomic and molecular collision cross sections throughout not only the classical Langevin regime, but the entirety of the collision energy domain. − And, others have revived attention on unique atomic systems in which the quantum-to-classical crossover not only appears in the ultracold regime, but in fact directly at zero Kelvin, which entails classical and semiclassical scaling laws that hold at all collision energies, as confirmed by both simulation and experiment. ,,,,,,− ,,,−

2.3.1. Kinematic Derivations

Today’s derivations of semiclassical scaling laws and holistic quantum models for ultracold collisions can largely be divided into two strategies: kinematic derivations, , which consider the geometry of phase space, and dynamical derivations, ,,, which consider the fate of individual particles.

Today’s kinematic derivations of semiclassical and classical post-threshold laws , build on the foundation laid by Wigner over 75 years ago in his original quantum threshold law: the idea that it is possible to derive (quasi)­universal laws for cross section scaling using only the form of the long-range potential and the system’s dimensionalityall with no knowledge of the exact short-range forces between collision partners.

A prime example is the derivation of a quasiuniversal post-threshold law for cold and ultracold dipolar scattering of Bohn, Cavagnero, and Ticknor in 2009. Based on prior kinematic quantum threshold law derivations ,− and the theoretical low-energy molecular collision studies of Kajita; ,− DeMille, Glenn, and Petricka; and Roudnev and Cavagnero; , Bohn, Cavagnero, and Ticknor employed a kinematic approach to develop a quasiuniversal post-threshold law for cold and ultracold dipolar scattering based on the eikonal approximation. This derivation rested on calculation of the total scattering cross section arising from an approximate eikonal wave function in a dipole–dipole potential in the presence of an applied field. They assumed that the wave function amplitude in the interaction region varies slowly and is therefore amenable to the cylindrical-coordinate phase-amplitude Ansatz

$$\psi (\mathrm{r⃗})={\mathrm{e}}^{\mathrm{i}{\mathrm{k⃗}}_{\mathrm{i}}·\mathrm{r⃗}}\exp [-\frac{\mathrm{i}}{k}{\int }^{z}V(b,\varphi ,z^{\prime })\mathrm{d}{z}^{\prime }]$$ 34

for initial, final, and average wavenumber k⃗ _i, k⃗ _f, and k⃗ _avg = (k⃗ _i + k⃗ _f)/2, respectively. For a dipole–dipole potential

$$V(b,\varphi ,z)=\frac{1}{{(b^2+z^2)}^{3/2}}[1-3\frac{{(\mathrm{b⃗}·\mathcal{Ê}+z{\mathrm{k̂}}_{\mathrm{avg}}·\mathcal{Ê})}^2}{b^2+z^2}]$$ 35

in an applied field

$$\mathcal{Ê}=\sin \alpha \cos \beta \mathrm{x̂}+\sin \alpha \sin \beta ŷ+\cos \alpha {\mathrm{k̂}}_{\mathrm{avg}}$$ 36

$$\alpha =\arccos (\mathrm{k̂}·\mathcal{Ê}),\beta =\arctan (ŷ·\mathcal{Ê}/\mathrm{x̂}·\mathcal{Ê})$$ 37

they derived the eikonal integral scattering amplitude as

$$\begin{array}{ll}{f}^{\mathrm{Ei}}& =\frac{k}{2\pi \mathrm{i}}\int b\mathrm{d}b\mathrm{d}\varphi {\mathrm{e}}^{\mathrm{i}qb\cos \varphi }\\ & \times [\exp \{\mathrm{i}\frac{2}{k{b}^2}{\sin }^2\alpha \cos (2\varphi -2\beta )\}-1]\end{array}$$ 38

Consideration of the optical theorem and averaging over initial wavevectors then gave the quasiuniversal semiclassical post-threshold law for dipolar scattering

$$\frac{{\sigma }_{\mathrm{Ei}}}{D^2}=\frac{8\pi }{3k}$$ 39

where D = Mμ ²/ℏ ² is the dipole length as a function of the reduced mass M and dipole moment μ of the dipole in question.

Concurrently, Hudson employed the same strategy to develop an eikonal approximation to the total elastic scattering cross section for molecular ion collisions of velocity v subject to an interaction potential of the form −C ₄/r ⁴ 

$${\sigma }_{\mathrm{Eik}}=\mathrm{\Gamma }\left(\frac{1}{3}\right){\left(\frac{\pi C_4}{2\hslash v}\right)}^{2/3}$$ 40

and Ticknor developed an eikonal approximation to the total cross section for dipolar molecules confined to two dimensions ,

$${\sigma }_{\mathrm{SC}}=\frac{4}{k}\sqrt{\pi Dk}$$ 41

In 2024, Wang and Bohn followed with a similarly eikonal approximation to the total cross section for microwave-shielded collisions of distinguishable dipoles

$${\mathrm{\sigma ̅}}_{\mathrm{total}}^{\mathrm{Ei}}=\frac{8\pi a_d}{3k}$$ 42

for dipole length a _d . This eikonal scaling derivation resulted in complete agreement with Bohn, Cavagnero, and Ticknor’s prior point-dipole universal semiclassical scaling law.

2.3.2. Dynamical Derivations

A greater number of recent universal semiclassical scaling laws ,,, pull from the dynamical tradition founded by Langevin and Gorin: − ,, that scaling laws for classical capture can be wholly determined by whether classical particles of interest would have sufficient radial momentum to surmount interposing centrifugal barriers.

In the early 2010s, Quéméner and Bohn − and Gao − recognized that the dynamical classical capture approach holds post-threshold for ultracold collisions. This insight served as the basis for their development of holistic quantum models based on classical input that reproduce the expected scaling laws in both the threshold and post-threshold regimes. Quéméner and Bohn developed the quantum threshold model, which replicates both threshold and post-threshold scaling behavior by assuming the quenching probability follows the Bethe–Wigner threshold law below the barrier and the Langevin (post-threshold) model at or above the barrier − ,

$${|T_{L,M_L}^{\mathrm{qu}}|}^2=\{\begin{array}{ll}{(E_c/V_b)}^{L+1/2}& E_c<V_b\\ 1& E_c\ge V_b\end{array}$$ 43

Meanwhile, Gao developed the quantum Langevin model, , which assumes the submatrix S̃ _eff of the S matrix in multichannel quantum defect theory is approximately zeronamely, it is unlikely for molecules to exit their initial configurations upon close approachin agreement with the Langevin assumption in a quantum context. The dynamical Langevin classical capture approach has also been used for purposes beyond cross section determination for derived scaling laws that have been shown to hold in a cold and ultracold atomic and molecular context, such as Pinkas et al.’s 2023 use of classical Langevin capture to determine the probability of Langevin collisions relevant to the generation of ⁸⁸Sr⁺ – ⁸⁷Rb ion–atom molecules in a linear Paul trap and Kłos and Tiesinga’s 2023 use of a semiclassical approximation to determine both semiclassical glancing rate coefficients and thermalized total elastic rate coefficients, the latter to within ∼10 to 20% accuracy of quantum mechanics for ultracold ⁷Li and ⁸⁷Rb collisions with room-temperature H₂, ⁴He, ²⁰Ne, ¹⁴N₂, ⁴⁰Ar, ⁸⁴Kr, and ¹³²Xe.

In 2018, ref presented ultracold inferno theory as another approach to dynamically determine classical post-threshold laws in ultracold chemistry. Ref argued that classical phase-space counting is sufficient to determine the post-threshold law for collision complex decay in a simple, two-dimensional model of ultracold chemical reactions as long as there is no quantum interference and the reaction is sufficiently exothermic to preclude quantum reflection along the reaction coordinate. Derivation of the post-threshold law then boils down to determination of a phase-space bottleneck: the minimum radial momentum at a given radius that leads to collision complex decay, beyond which any additional angular momentum leads to reformation of the collision complex. Based on this phase-space bottleneck insight, the law was determined for nonpolar homonuclear diatomic point particles subject to a central van der Waals potential

$$V(r)=-C_6/{(\beta r^2+\alpha )}^3$$ 44

with dispersion coefficient C ₆, parametrization constants α and β, and chaotic motion modeled by energy-preserving phase-space kicks within radius r ≤ R. ,, In direct analogy to classical capture, the approach indicated that a classical particle of mass m that surmounts the effective potential energy maximum

$$E_{\mathrm{crit}}=\frac{{\beta }^{3/2}{|p_{\theta }|}^3}{3^{3/2}{2}^{1/2}{C}_6^{1/2}{m}^{3/2}}$$ 45

at critical radius

$$r_{\mathrm{crit}}={\left(\frac{6m{C}_6}{p_{\theta }^2{\beta }^3}\right)}^{1/4}$$ 46

and the critical momenta

$$p_{\theta ,\mathrm{crit}}=3^{1/2}{2}^{1/6}{C}_6^{1/6}{m}^{1/2}{E}^{1/3}{\beta }^{-1/2}$$ 47

$$p_{r,\mathrm{crit}}={(2mE+2m{C}_6{(\beta r^2+\alpha )}^{-3}-p_{\theta ,\mathrm{crit}}^2{r}^{-2})}^{1/2}$$ 48

will pass unimpeded to r → ∞ and lead to collision complex decay. In contrast, particles that cannot reach the critical radius r _crit can pass no further than the classical turning point radius

$$R_{\mathrm{turn}}={\left(\frac{2C_6}{E{\beta }^3}\right)}^{1/6}$$ 49

Alternatively, in Birkhoff coordinates (s,p _s ) defined at a fixed-radius annulus r = R _Birk around the origin, the critical momentum normal to the annulus is

$$p_{s,\mathrm{crit}}=\sin \left[\arctan \left(\frac{p_{\theta ,\mathrm{crit}}}{R_{\mathrm{Birk}}{p}_{r,\mathrm{crit}}}\right)\right]$$ 50

such that the phase-space bottleneck and angle of acceptance is simply

$$r=R,|p_s|<p_{s,\mathrm{crit}}$$ 51

According to (semi)­classical phase-space counting, the ergodic rate given by the ratio of the volume flux for escaping particles to the total phase-space volume of nonescaping particles is then

$$k(E)=\frac{4p_{\theta ,\mathrm{crit}}}{m\pi R_{\mathrm{turn}}^2-4mF(R,R_{\mathrm{turn}})}$$ 52

$$F(a,b)={\int }_a^b\mathrm{d}rr\arctan \left(\frac{p_{\theta ,\mathrm{crit}}}{r{p}_{r,\mathrm{crit}}}\right)$$ 53

as a function of the turning-point radius R _turn. The resulting rate scales as k(E) ∝ E ^2/3, in close agreement with the results of classical trajectory simulations.

2.3.3. (Semi)­classical Analog of Fermi’s Golden Rule

In light of these classical successes, it is worthwhile to reconsider Mody et al.’s 2001 approach to derive (semi)­classical post-threshold laws for today’s ultracold atomic and molecular collisions in alternative (e.g., reduced) dimensionalities. Mody et al.’s approach, which can be interpreted both kinematically and dynamically, recognized that where accurate semiclassical wave functions can be used to approximate the wavenumber dependence of the transition rate Γ _ij according to a Fermi’s golden rule approach ,

$${\mathrm{\Gamma }}_{fi}=\frac{2\pi }{\hslash }{|V_{fi}|}^2\rho (E_f)$$ 54

given a semiclassical density of states ρ­(E _f ) and matrix element V _fi . The post-threshold rate law then reduces to knowledge of the scalings of the probability density and the density of states as a function of the collision energy E _f . Originally introduced by Mody et al. in 2001 to describe three-dimensional two-body collisions, in the presentation that follows, this review generalizes the approach to N dimensions in order to enable the technique to describe both the full-dimensional and reduced-dimensional two-body ultracold collisions currently underway experimentally, as follows: According to the semiclassical Weyl law , and kinematics, the number of states accessible in a given region of phase space Ω at energy E for a Hamiltonian $\mathcal{H}$ in N dimensions is

$$n(E)={\iint }_{\mathrm{\Omega }}\mathrm{\Theta }(E-\mathcal{H})\mathrm{d}\mathbf{p}\mathrm{d}\mathbf{x}$$ 55

$$\propto k^N$$ 56

where the Heaviside step function is

$$\mathrm{\Theta }(a)=\{\begin{array}{ll}0& a<0\\ 1& a\ge 0\end{array}$$ 57

and the momentum p(E) is approximated as ℏk in the asymptotic region. The gradient of the number of states with respect to energy then yields the semiclassical density of states in the absence of an electromagnetic field interaction

$$\rho (E_f)\propto k^{N-2}$$ 58

In the absence of tunneling, the squared average overlap ${\stackrel{\circledR }{|{\psi }_f{\psi }_i|}}^2,$ which determines the wavenumber-dependence of the average matrix element squared V _fi , is proportional to the particle probability densities |ψ _i |² and |ψ _f |². The probability density of the incoming wave |i⟩ scales as k, according to unit flux normalization. However, the probability density of the scattered wave |f⟩ is not as straightforward: Here, the classical particle probability density is inversely proportional to the allowed volume in phase space. To illustrate, the amount of phase space accessible in the inner region at a given energy is pictured in Figure . As the dimensionality increases, the amount of accessible phase space also increases, which leads to a drop in the particle probability density. The semiclassical Weyl law gives the scaling of this reduction as

$${|\psi (x)|}^2\propto k^{-(N-2)}$$ 59

19.

Classical particle probability density |ψ­(x)|² eq declines as the dimension of the system increases due to a concomitant increase in the extent of energetically accessible phase space, as depicted for a Lennard-Jones potential in one dimension (|ψ­(x)|² ∝ k ^–1), two dimensions (|ψ­(x,y)|² ∝ k ⁰), and three dimensions (|ψ­(x,y,z)|² ∝ k ¹).

When multiplied with the expression for the density of states, the density of states and matrix element wavenumber dependence combine to yield the semiclassical post-threshold law

$$\sigma \propto 1$$ 60

$$\mathcal{R}\propto k$$ 61

This universal semiclassical scaling law has an intuitive universal, classical interpretation in line with a dynamical classical capture theory approach: in the absence of dynamical barriers, particles with sufficient energy to surmount any intervening potential barriers separate.

2.4. Classical Behavior at Zero Kelvin

Authors have identified two key classes of systems that exhibit classical scaling laws directly at zero Kelvin: scattering from attractive Coulomb potentials in three dimensions ,,,,,,− and multiple ionization. ,,− Arguments for this extreme classical behavior at infinite wavelength pull from both the semiclassical Wentzel–Kramers–Brillouin criterion , or the Van Vleck–Morette–Gutzwiller propagator, , as outlined below.

2.4.1. Scattering from Attractive Coulomb Potentials in Three Dimensions

Modern inquiry into scattering from attractive Coulomb potentials in an ultracold atomic context arose from a controversy surrounding atom–surface sticking coefficients and atom–grating reflection (refs , , − , − , , and − ): whether the low-energy limit of the sticking coefficient for hydrogen atoms on superfluid ⁴He follows the expected quantum threshold law s(T) ∝ T ^1/2, reaches a constant value reminiscent of the classical result s(T) ∝ 1, and/or succumbs to decoherence and resonance effects that alter the threshold law. A lasting legacy of this inquiry has been the recognition that classical and semiclassical mechanics yield not only approximate, but exact quantum scattering cross sections for three-dimensional attractive Coulomb potentials at all energies.

Several arguments explain the origin of this all-energy quantum-classical correspondence. ,,, The early argument of Clougherty and Kohn in 1992 recognized that the sticking coefficient for attractive Coulomb tails z ^–1 scales as the s(E) ∝ α, as initially observed in atom–surface sticking experiments, in contrast to the standard quantum threshold law s(E) ∝ E ^1/2 identified for attractive z ^–3 potential tails identified by Böheim, Brenig, and Stutzki. Clougherty and Kohn attributed the discrepancy to the fact that Coulomb potentials always satisfy the semiclassical WKB criterion for sufficiently large radii z where van der Waals potentials of order β > 2 do not, as follows: where the WKB criterion for a van der Waals potentials of the form U(z) = C _β /z ^β is

$$k^2(z)\gg \left|\frac{\mathrm{d}k(z)}{\mathrm{d}z}\right|$$ 62

$$1\gg \frac{\beta C_{\beta }}{2z^{\beta +1}{(k^2+\frac{C_{\beta }}{z^{\beta }})}^{3/2}}$$ 63

where k ² (z) ≡ k ² + U(z), Clougherty and Kohn recognized that for short-range potential with van der Waals component β > 2, semiclassical results never hold as z → ∞. As an example, they demonstrated that β = 3 satisfies the WKB criterion taken as an equality when

$$1=\frac{3C_3}{2{[z^{*}(k)]}^4{(k^2+\frac{C_3}{{[z^{*}(k)]}^3})}^{3/2}}$$ 64

$$z^{*}(k)={\left(\frac{3C_3}{2}\right)}^{1/4}{k}^{-3/4}$$ 65

In this case, $\underset{k\to 0}{\lim }{z}^{*}(k)=\infty$ , quantum reflection appears at arbitrarily distant radii z** ∼ γk ^–1 for γ ≪ 1, and semiclassical and quantum results do not agree. In contrast, Clougherty and Kohn demonstrated that the Coulomb potential β = 1 satisfies the WKB criterion taken as an equality when

$$1=\frac{C_1}{2{[z^{*}(k)]}^2{(k^2+\frac{C_1}{z^{*}})}^{3/2}}$$ 66

$$z^{*}(0)=\frac{1}{4C_1}$$ 67

$$z^{*}(k)<z^{*}(0)$$ 68

The WKB criterion then holds for all k for sufficiently large radii z > z**, z** ≫ z* (0) has no k-dependence, and semiclassical and quantum results agree.

In 2001, Mody et al. also put forward a WKB argument for the semiclassical nature of attractive Coulomb potential scattering. This WKB argument relied on showing that, although the maximal WKB criterion diverges in the threshold limit for van der Waals potentials of order n > 2, the maximal WKB criterion remains small near threshold for van der Waals potentials of lower order n ≤ 2 in the asymptotic regime. Mody et al. recognized that the WKB criterion

$$|{\lambda }^{\prime }(x)|=\left|\frac{\hslash p^{\prime }}{p^2}\right|\ll 1$$ 69

is maximal where

$$\frac{p^2}{3m}=\frac{V^{\prime 2}}{V^{″}}$$ 70

They determined that the maximum WKB criterion for a van der Waals potential V(x) = −c _n /x ⁿ for n > 2 scales as

$$\max \left|\frac{p^{\prime }}{p^2}\right|\sim \frac{1}{c_n^{1/n}{\epsilon }^{1/2-1/n}}\sim \frac{1}{c_n^{1/n}{k}^{1-2/n}}$$ 71

which approaches infinity as k → 0indicative of WKB failure. Mody et al. then showed that, in contrast, the maximal WKB criterion remains small at all radii sufficiently far from the origin for van der Waals potential n ≤ 2: the WKB approximation then holds at sufficiently large radii for Coulomb potentials, in direct agreement with Clougherty and Kohn.

Yet, in arguments laid out from 2004 to 2017, Friedrich and Trost noted several properties that raise concerns with such WKB arguments. ,,, Although the WKB approximation holds at sufficiently large radii for attractive Coulomb potentials, the WKB approximation still fails close to the origin. Arguments based on the WKB approximation also do not contend with the fact that, although quantum and classical scattering cross sections agree in three-dimensional Coulomb potentials, they do not agree in two-dimensional Coulomb potentials, as identified independently by Barton in 1983 and Lin in 1997. Finally, the focus on specific radii associated with transitions in quantum reflection behavior (such as z ^*) also appear counter to the Coulomb potential’s property of mechanical similarity for dimension d = −1 and scaling factor a

$$V_d(ax)=a^d{V}_d(r)$$ 72

In such mechanically similar systems, for any classical path r(t) of energy E in a Coulomb potential, there must be a valid classical path ar(r ^3/2 t) of energy E/a. Thus, each radius is in a loose sense on an equal footing.

An argument based on the Van Vleck–Morette–Gutzwiller propagator, introduced in 1994 and revisited in light of Friedrich and Trost’s findings in 2019, avoids these pitfalls by recognizing that a three-dimensional attractive Coulomb potential obeys a classical scaling law because of a hidden equivalence to another system known to behave classically: , a regularization followed by a canonical transformation metamorphosizes a three-dimensional Coulomb potential into a collection of independent harmonic oscillators. In detail, the Hamiltonian for a three-dimensional Coulomb potential

$$h=\frac{{\mathbf{p}}^2}{2m}-\frac{Z{e}^2}{r}$$ 73

upon Kustaanheimo–Stiefel regularization with positions and momenta

$$x_1=X_1^2-X_2^2$$ 74

$$x_2=2X_1{X}_2$$ 75

$$x_3=X_3^2-X_4^2$$ 76

$$x_4=2X_3{X}_4$$ 77

$$P_i=\sum _{j=1,4}{p}_j\frac{\partial x_j}{\partial X_i}$$ 78

$$r=\sqrt{\sum _{i=1}^4{x}_i^2}=\sum _{i=1}^4{X}_i^2$$ 79

$$\mathrm{d}\tau =\frac{\mathrm{d}t}{r}$$ 80

yields the Hamiltonian

$$H=\sum _{i=1}^4(\frac{P_i^2}{8m}-E{X}_i^2)+Z{e}^2$$ 81

which is none other than a system of four uncoupled harmonic oscillators. Note the kinetic energy operator remains unchanged such that no changes are needed to ensure position and momenta remain canonical coordinates. The quantum Hamiltonian then follows directly as

$$[\sum _{i=1}^4(-\frac{{\hslash }^2}{8m}\frac{{\partial }^2}{\partial X_i^2}-E{X}_i^2)+Z{e}^2]\mathrm{\Psi }=0$$ 82

and the quantum scattering amplitude may be determined via the pseudotime quantum propagator. What remains is to show agreement between the pseudotime quantum propagator and the classical propagator. Since the classical action of harmonic oscillators is quadratic, the propagator is semiclassically exact. And, since the system features no quantum interference, semiclassical and classical results agree. Thus, attractive Coulomb scattering in three dimensions features quantum-classical correspondence at all energiesincluding directly at zero Kelvin. ,

2.4.2. Wannier Threshold Law, Multiple Ionization, Atomic Fragmentation, and Implications for Ultracold Chemistry

Advancements in atomic and molecular cooling have also led to a return to ,,,,, Wannier’s threshold laws for double and multiple , ionizationspecifically regarding the source of the renowned ability of the Wannier threshold law for double ionization to hold directly at zero Kelvin, as has been confirmed semiclassically, − quantum mechanically, − ,− and experimentally to at least three digits in the argument of the exponential. − (Note some dissent in favor of a linear scaling law in recent years, , as well as the presence of a high-profile, fully quantum dynamics study of the system, which has no reference to Wannier.)

In 2019, it was put forward that a possible explanation of why classical mechanics is sufficient to predict the threshold law for double atomic ionization follows from a generalization of the aforementioned Van Vleck–Morette–Gutzwiller argument given for the accuracy of classical threshold laws in attractive three-dimensional Coulomb potentials. Analysis of the potential energy associated with two electrons escaping a central atom

$$V=-\frac{Z{e}^2}{r_1}-\frac{Z{e}^2}{r_2}+\frac{e^2}{r_{12}}$$ 83

indicates that the potential energy for double ionization reduces to a single Coulomb potential in the limit of double electron escape r ₁ = – r ₂, r ₁₂ → ∞, which implies that the Kustaanheimo-Stiefel regularization again allows the Hamiltonian to be expressed as a set of harmonic oscillators in the asymptotic limit. The pseudotime propagator is then semiclassically exact and semiclassical and quantum cross sections agree. Likewise, as in Rutherford scattering, only one path between initial and final states contributes to double ionization, such that there is no quantum interference and semiclassical and classical cross sections agree. Quantum and classical cross sections then agree at all energies, including zero Kelvin.

The same work introduces an important caveat: although the Hamiltonian for double ionization can be expressed as a sum over harmonic oscillators in the asymptotic region, in the inner region, interaction between the electrons obviates such an expression. This distinction can be readily confirmed by analysis of the form of the transformed potential energy away from asymtopia, with the collinear case as an example. In the collinear case, − the regularized Hamiltonian, given by two Kustaanheimo-Stiefel and one Levi-Civita − regularizations, is

$$H=\frac{P_1^2{R}_2^2}{8}+\frac{P_2^2{R}_1^2}{8}-Z{R}_1^2-Z{R}_2^2+R_1^2{R}_2^2+\frac{R_1^2{R}_2^2}{R_{12}^2}$$ 84

This expression contains nonquadratic terms, such that the propagator is not expected to be semiclassically exact in the inner region.

Although not recently discussed in the field of ultracold atomic and molecular collisions, the success of Wannier’s classical threshold law gave rise to classical scaling laws for multiple (n-fold) ionization that has continued to attract some interest in near-threshold physics in recent years. ,− Wannier put forward a classical approach to determine the threshold law for multiple ionization in a half-page article in Physical Review, which Geltmann confirmed quantum mechanically: the threshold laws for ionization by electrons and photons are σ­(E) ∝ E ⁿ and σ­(E) ∝ E ^n–1, respectively. Subsequent generalizations of Wannier’s threshold law include development and verification of threshold laws for triple ionization, ,− ,− quadruple ionization, quintuple ionization, and generic multiple ionization − and atomic fragmentation, , which inspire investigation of such behaviors in an ultracold context.

2.5. Summary of Universal (Semi)­classical Near-Threshold Scaling Laws

In order to highlight where (semi)­classical post-threshold laws are expected to be accurate in cold and ultracold collisionsand where classical mechanics is expected to give accurate cross section and rate scalings for all energiesthis review briefly summarizes key universal quantum threshold and classical post-threshold laws in one place, Table . These universal scaling laws are expected to apply regardless of the species of the scatterers in question, including in reduced dimensional scattering systems which are gaining in interest since 2010. ,− However, as reflected in recent work, ,− ,− cross section scalings may deviate from expected scalings. Newly open channels may lead to Wigner cusps. ,− Bound states or resonances may abbreviate or alter the s-wave quantum threshold regime, as studied in-depth in recent years by Macri, Barrachina et al. − and Simbotin, Côté, and Ghosal. , And, other near-threshold phenomena may intervene, affecting the near-threshold scaling laws, including threshold resonances, proximity resonances, ,, the Thomas effect, − the Efimov effect, ,− ,− Borromean binding, and Brunnian binding. − Nonetheless, the independence of the laws on the short-range form of the potentialand the exact strength of the asymptotic potentialsuggest the quantum-classical correspondences predicted therein are expected to be even more widespread. ,

7. Summary of Universal (Semi)­classical Scaling Laws for Two-Body s-Wave Endothermic Collisions Subject to Short-Range Potentials in Varied Dimensionality (1D to ND), the Attractive Coulomb Potential in Three Dimensions (Coulomb), and the Wannier Double Ionization System in Full Dimensionality (Wannier) .

^aBoth (semi)­classical (class) and quantum (quant) scalings are provided for the probability density |ψ|², cross section σ, and rate $\mathcal{R}$ in terms of the wavenumber k, with (semi)­classical scalings also provided for the density of states ρ _class. The semiclassical 1D-ND and Coulomb scaling laws are determined according to the classical analogue of the Fermi Golden Rule, , with classical scaling laws determined for the Wannier system according to phase-space counting. Quantum threshold laws (quant) are included for direct comparison, with instances of quantum-classical correspondence bolded and emphasized in blue.

3. Classical and Quasiclassical Methods for Ultracold Molecular Collision Complex Decay

Classical and quasiclassical methods have now taken center stage in what has been termed the ultracold “sticky mystery”: In the 2010s, ground-state alkali metal dimer collision experiments observed reactant loss not only for exothermic ultracold chemical reactions, where disappearance of reactants is expected, but also for nonreactive ultracold collisions, where no products are energetically favorable to form. − Mayle et al. hypothesized this reactant loss was not due to product formation, but instead the formation of long-lived (namely, “sticky”) collision complexesa hypothesis that was confirmed experimentally for the ultracold KRb dimer reaction via the direct measurement of the ${{}^{40}\mathrm{K}}_2^{87}{\mathrm{Rb}}_2^{*}$ collision complex by Hu et al.

The question then remained: just how long-lived are such collision complexes? Investigation of such collision complexes remains computationally intractable with today’s exact quantum dynamical methods. The low energies of product moleculesat times on the order of 100 nKare associated with extremely long wavelengths, which necessitate simulation of broad swaths of position space. As Bause et al. stated in 2023, formation of the collision complex can be energetically favorable by on the order of 1000 K, such that on the order of 10,000 rotational states may be required to represent just one dimer molecule alone. And, treatment of a four-atom alkali metal dimer collision complex constitutes a six-dimensional problem, which lies at the upper limit of the dimensionality amenable to standard grid-based quantum dynamics methods for relatively light atoms at high temperatures. Croft, Balakrishnan, and Kendrick put this cost into context in 2017:

• Quantum-scattering calculations for collisions of ultracold molecules are extremely computationally expensive. Even...for only total angular momentum J = 0, even exchange symmetry and parity, [and without inclusion of] spin or field effects...calculations [of scattering cross sections in the ultracold K₂ + Rb collision] still required over 300,000 hours of CPU time. It is possible that numerically exact quantum scattering calculations including all the effects omitted in these calculation will never be computationally tractable.

This extreme computational cost led to the development of computationally tractable alternatives based on classical, semiclassical, and quasiclassical mechanics. −

3.1. Quantum Rice–Ramsperger–Kassel–Marcus Theory Approach

In 2012 and 2013, Mayle et al. , laid the foundations for classical approaches to collision complexes in ultracold collisions when they recognized that long-lived collision complexes correspond to both high numbers of Fano-Feshbach resonances associated with rovibrational states quantum mechanically and winding trajectories that ergodically sample phase space classically, such that the collision complex lifetime can be determined classically or quasiclassically notwithstanding the ultracold temperature of the initial scattering partners (see Figure ). This statistical behavior allowed the collision complex to be determined according to Rice–Ramsperger–Kassel–Marcus (RRKM) theory, − which models unimolecular dissociation as a process in which energy passes throughout all degrees of freedom until it concentrates on a single mode driving dissociation. Calculation of the collision complex lifetime according to RRKM theory

$${\tau }_{\mathrm{RRKM}}=\frac{2\pi \hslash \rho }{N_o}$$ 85

then requires knowledge of only the number of open scattering channels N _o and the collision complex density of states ρ. In 2012 and 2013, Mayle et al. , estimated the required density of states quantum mechanically by calculating the bound state energies E _α of an approximate potential energy surface to determine the short-range resonances for vibrational quantum number α, offsetting the short-range resonances by asymptotic rovibrational channel (v,n) threshold energies to determine the rovibrational channel short-range resonance states E _α = E _α + E _{v ,n} , analyzing the E _α spectrum to determine the mean level spacing d, and employing the mean level spacing to determine the rovibrational density of states ρ_rv = 1/d. Mayle et al. found that quantum RRKM theory yielded collision complex lifetimes for both atom–diatom and diatom–diatom collisions that were consistent with the “sticky mystery” observation of reactant loss in both reactive and nonreactive ultracold collisions. In addition, Mayle et al. estimated semiclassically the likelihood of hyperfine state changes during the collision process, and noted that the density of states is unaffected by the number of nuclear spin states N _nuc, as N _o and ρ scale according to the same multiplicative factor τ = 2πℏρ/N _o = 2πℏ(ρ_rv N _nuc)/(N _o N _nuc) = 2πℏρ_rv. This approach has been yet further extended by Croft, Bohn, and Quéméner in the early 2020s via the incorporation of transition-state-theory and optical model corrections that improve the magnitude, if not the energy scaling, of the predicted time dependence of the relative population of the collision complex. ,

20.

Current picture of sticky ultracold metal dimer collisions, in which (bottom left) an initial state of molecules with a kinetic-energy distribution of temperature T passes through a centrifugal collision threshold to form (bottom center) a collision complex that supports a large density of states. The collision complex behaves like (top left) a classical particle that samples phase space ergodically until reaching a configuration compatible with formation of the product dimers, either via (top right) photoexcitation to the excited complex and spontaneous emission or predissociation or an unknown process leading to (bottom right) molecule loss. Reproduced from ref . Copyright 2023 American Chemical Society.

3.2. Classical Trajectory Approach

In 2014, Croft and Bohn recognized that classical trajectories can also yield collision complex lifetimes on par with quantum RRKM theory. Croft and Bohn simulated classical homonuclear atom–diatom collision trajectories and recorded the time required in each trajectory for the atom to reach a distance from the diatom R _∞ with sufficient momentum to escape irreversibly. The fraction of atoms remaining in the complex at time t followed an exponential distribution

$$f=\exp (-\frac{t}{{\mathrm{\tau ̅}}_{\mathrm{ct}}})$$ 86

as expected for ergodic trajectories, which supported the underlying ergodicity assumption of prior RRKM treatments. Moreover, the collision complex lifetimes τ̅ _ct agreed to within an order of magnitude of quantum RRKM theory lifetimes for Li + Li₂, K + K₂, and Cs + Cs₂ (see Table ), which opened the door to subsequent quasiclassical treatment of collision complex lifetimes. Classical trajectory results for lifetimes in agreement with RRKM theory were also obtained for K + K + Rb by Lide in 2021 and for NaK + K, NaK + NaK, and RbCs + RbCs, by Man, Groenenboom, and Karman in 2022, with additional collision complex lifetimes determined by Kłos et al. in 2021 with quasiclassical trajectories for ²³Na⁸⁷Rb + ²³Na⁸⁷Rb collisions evidencing roaming trajectories. Beyond collision complex lifetime prediction, Croft and Bohn also demonstrated that classical trajectories enabled investigation into the onset of chaos via calculation of the fractal dimension, which set the path to later identification of classical fractals in dipolar collisions by Yang, Pérez-Ríos, and Robicheaux in 2017.

8. Comparison of Collision Complex Lifetimes Determined via Quantum RRKM Theory τ _quant and Classical Trajectory Simulations τ̅ _ct for Triatomic Collision Systems .

System	τquant [ns]	τ̅ct [ns]
Li + Li2	2.4	12 ± 1
K + K2	146	303 ± 36
Cs + Cs2	2746	2871 ± 328

^aThe quantum and classical lifetimes agree to within one standard deviation for Cs + Cs₂, with greater deviation observed for Li + Li₂ and Cs + Cs₂. Adapted table with permission from James F. E. Croft and John L. Bohn, Physical Review A, 89, 1, 012714, 2014. Copyright 2014 by the American Physical Society.

It is worthwhile to note that such classical trajectory approaches to ultracold collisions extend beyond sticky collisions. Classical trajectories have also been used to examine the possible formation of collision complexes in buffer gas cooling of low-energy complex hydrocarbons in 2012 and 2014 , and to identify the role of quasiresonant vibration–rotation energy transfer in low-energy atom–diatom collisions by Forrey et al. in 1999 and 2001. , For example, in quasiresonant transfer, a collision time that exceeds the rotational period can lead to a surprising specificity in the transfer of energy between vibrational and rotational degrees of freedom: namely, adiabaticity of the action I leads to dominance of a highly specific set of collisions that satisfy

$$\mathrm{\Delta }I=n_v\mathrm{\Delta }v+n_j\mathrm{\Delta }j=0$$ 87

where Δv and Δj are the changes in the vibrational and rotational quantum numbers, respectively, and where n _v and n _j satisfy the quasiresonant condition

$$\frac{n_v}{n_j}\approx \frac{{\omega }_v}{{\omega }_j}$$ 88

where ω _v and ω _j are the classical vibrational and rotational frequencies, respectively. In 1999, Forrey et al. employed classical trajectories to investigate whether classical quasiresonant transfer extends into the cold regime, and showed that ultracold ⁴He – H₂ collisions of collision energy E _i = 10^–6 au (0.32 K) with initial quantum numbers (v _i , j _i ) = (2,21) hewed closely to Δj = −2Δv and a quasiresonant condition ω _v /ω _j ≈ 2.079.

3.3. Quasiclassical Rice–Ramsperger–Kassel–Marcus Theory Approach

In 2019, Christianen, Karman, and Groenenboom built on these results to demonstrate that quasiclassical RRKM theory also allows for accurate determination of collision complex lifetimes. This advancement was essential to overcome difficulties extending previous quantum RRKM theory approaches to collision complex lifetimes beyond isotropic, bond-length-independent potentials. Christianen, Karman, and Groenenboom employed a quasiclassical phase-space counting approach to determine the density of states, and thereby the RRKM rate eq : they approximated the number of quantum states supported by an N-atom collision complex of N _i indistinguishable particles of identity i with total angular momentum J ₀ at energies up to E

$$\begin{array}{ll}{N}^{(\mathrm{cl})}(E,{\mathbf{J}}_0)& =\frac{1}{3^{3N-3}\prod _i{N}_i!}\int \mathrm{d}\mathbf{x}\int \mathrm{d}\mathbf{p}\mathrm{\Theta }[E-H(\mathbf{x},\mathbf{p})]\\ & \times \delta [\mathbf{P}(\mathbf{p})]\delta [\mathbf{X}(\mathbf{x})]\delta [{\mathbf{J}}_0-\mathbf{J}(\mathbf{x},\mathbf{p})]\end{array}$$ 89

where Θ­(E) is the Heaviside function, and evaluated the density of states as

$${\rho }^{(\mathrm{cl})}(E,{\mathbf{J}}_0)=\frac{\mathrm{d}{N}^{(\mathrm{cl})}}{\mathrm{d}E}$$ 90

The density of space for specific quantum numbers of the total angular momentum J, its projection M _J , and parity p is then given by

$${\rho }_{JMp}(E)=g_{\mathbf{N}Jp}{\int }_{JM}{\rho }^{(\mathrm{cl})}(E,\mathbf{J})\mathrm{d}\mathbf{J}$$ 91

where g _NJp is the proportion of classical phase space for quantum numbers N and J associated with parity p. Christianen, Karman, and Groenenboom subsequently employed analytic integration over the momentum coordinates to yield the three-atom AB + C field-free density of states

$$\begin{array}{ll}{\rho }_{\mathbf{N},Jp}^{(\mathrm{AB}+\mathrm{C})}(E)& =\frac{g_{\mathbf{N}Jp}4\sqrt{2}\pi (2J+1)m_{\mathrm{A}}{m}_{\mathrm{B}}{m}_{\mathrm{C}}}{h^3(m_{\mathrm{A}}+m_{\mathrm{B}}+m_{\mathrm{C}})g_{\mathrm{ABC}}}\\ & \times \int \frac{Rr}{\sqrt{\mu R^2+{\mu }_{\mathrm{AB}}{r}^2}}{[E-V(\mathbf{q})]}^{1/2}\mathrm{d}R\mathrm{d}r\mathrm{d}\theta \end{array}$$ 92

where the degeneracy factor is g _ABC = ∏ _i N _i !, the three-body reduced mass is μ = (m _A + m _B)m _C /(m _A + m _B + m _C), and the diatom reduced mass is μ_AB = (m _A m _B) /(m _A + m _B + m _C); as well as the four-atom AB + CD density of states

$$\begin{array}{ll}{\rho }_{JMp}^{(\mathrm{AB}+\mathrm{CD})}& =\frac{g_{\mathbf{N}Jp}4{\pi }^6(2J+1)m_{\mathrm{A}}^3{m}_{\mathrm{B}}^3{m}_{\mathrm{C}}^3{m}_{\mathrm{D}}^3}{h^9{(m_{\mathrm{A}}+m_{\mathrm{B}}+m_{\mathrm{C}}+m_{\mathrm{D}})}^3{g}_{\mathrm{ABCD}}}\\ & \times \int \frac{R^4{r}_1^4{r}_2^4{\sin }^2({\theta }_1){\sin }^2({\theta }_2)}{\det \mathcal{I}(\mathbf{q})\sqrt{\det \mathcal{A}(\mathbf{q})}}{[E-V(\mathbf{q})]}^2\mathrm{d}\mathbf{q}\end{array}$$ 93

$$\begin{array}{ll}\mathcal{A}(\mathbf{q})& =\frac{1}{2}\sum _i[{\mathcal{K}}_i{(\mathbf{q})}^T{m}_i{\mathcal{K}}_i(\mathbf{q})\\ & -{\mathcal{D}}_i{(\mathbf{q})}^T{[{\mathcal{I}}^{(\mathrm{bf})}(\mathbf{q})]}^{-1}{\mathcal{D}}_i(\mathbf{q})]\end{array}$$ 94

$${[{\mathcal{K}}_i]}_{jk}=\partial {[{\mathbf{x}}_i]}_j/\partial {\mathbf{q}}_k$$ 95

$${[{\mathcal{D}}_i]}_{jk}=m_i\sum _{lm}{\epsilon }_{jlm}{[{\mathbf{x}}_i^{(\mathrm{bf})}(\mathbf{q})]}_l{[{\mathcal{K}}_i(\mathbf{q})]}_{mk}$$ 96

where $\mathcal{I}(\mathbf{q})$ is the inertial tensor, ε _jlm is the Levi-Civita tensor, and ^(bf) refers to body-fixed frame. Christianen, Karman, and Groenenboom also considered the case of external fields. Given an external field sufficient to cause nonconservation of the angular momentum quantum number J, the density of states becomes

$$\begin{array}{ll}\rho (E)& =\frac{16{\pi }^3{C}_{\mathbf{N}\mathbf{m}}}{\mathrm{\Gamma }(\frac{D}{2}+1)}\\ & \times \int \mathrm{d}\mathbf{q}G(q)\sqrt{\frac{\det \mathcal{I}(\mathbf{q})}{{\mathcal{I}}_{\mathrm{rot}}(\mathbf{q})}}{\{\pi [E-V(\mathbf{q})]\}}^{D/2}\end{array}$$ 97

where ${\mathcal{I}}_{\mathrm{rot}}$ represents the rotationally averaged ${\mathcal{I}}_{zz}$ . A second external field can also lead to nonconservation of the angular momentum quantum number M. The density of states then becomes

$$\begin{array}{ll}\rho (E)& =\frac{16\sqrt{2}{\pi }^3{C}_{\mathbf{N}\mathbf{m}}}{\mathrm{\Gamma }(\frac{D}{2}+\frac{3}{2})}\\ & \times \int \mathrm{d}\mathbf{q}G(\mathbf{q})\sqrt{\det \mathcal{I}(\mathbf{q})}{\{\pi [E-V(\mathbf{q})]\}}^{D/2+1/2}\end{array}$$ 98

A comparison of these quasiclassical density of states for field-free K₂ + Rb and NaK + NaK collisions indicated close agreement between quasiclassical and quantum results, as shown in Figure , which further supported quasiclassical treatment of the collision complex in the ultracold scattering systems. Given quasiclassical RRKM theory’s accuracy and versatility, quasiclassical RRKM theory is currently the state-of-the-art benchmark for experimental results, as exemplified by the work of authors such as Gregory et al., Liu et al., Bause et al., Gersema et al., and Nichols et al.

21.

Quantum (crosses) and quasiclassical (lines) density of states for ultracold metal dimer collisions closely agree, as demonstrated for K₂ + Rb (left) and NaK + NaK (right) both with (bottom) and without (top) applied fields. Dashed lines of slopes 3, 2.5, and 1.5 (left) and 5, 4.5, and 3.5 (right) indicate power-law scaling of the density of states, and dashed-dotted lines indicate quantum density of states for realistic potential energy surfaces not amenable to treatment with previous quantum RRKM theory approaches. Reproduced figure with permission from Arthur Christianen, Tijs Karman, and Gerrit C. Groenenboom, Physical Review A, 100, 3, 032708, 2019. Copyright 2019 by the American Physical Society.

Comparison between quasiclassical RRKM theory and experimental collision complex lifetimes in these works, as summarized in Table , reveals that quasiclassical RRKM theory successfully predicts collision complex lifetimes to within an order of magnitude in both the ⁸⁷Rb¹³³Cs + ⁸⁷Rb¹³³Cs and ⁴⁰K⁸⁷Rb + ⁴⁰K⁸⁷Rb collisions. ,, However, for ²³Na⁴⁰K + ²³Na⁴⁰K, ²³Na⁸⁷Rb + ²³Na⁸⁷Rb, ²³Na³⁹K + ²³Na³⁹K, and ⁴⁰K⁸⁷Rb + ⁸⁷Rb collisions, quasiclassical and experimental lifetimes differ by up to 5 orders of magnitude. ,, The source of this discrepancy remains an open question as of the writing of this review, with hypotheses ranging from the influence of conical intersections to the role of coupling between nuclear spin states.

9. Comparison of Quasiclassical τ _class and Experimental τ _exp Collision Complex Lifetimes of Ground-Hyperfine-State Alkali Metal Dimers/Atoms in Terms of an Order-of-Magnitude Multiplicative Factor .

^aSystems for which the quasiclassical and experimental lifetimes are of the same order of magnitude are emphasized in blue (see references [Refs.] and ref for additional details).

4. Outlook

The ongoing success of classical, semiclassical, and quasiclassical approaches to ultracold molecular collisions offers classical methods a position at the forefront of today’s quantum revolution. Of the variety of rationales given to explain the applicability of classical methods beyond the classical limit ℏ → 1, all can be applied to highlight the ability of classical, semiclassical, and quasiclassical mechanics to describe aspects of a wide variety of molecular systems deep in the prototypically quantum regime. ,,− ,,,, The rationales underline three key elements shared by ultracold molecular collisions amenable to such approaches: the presence of deep potential wells introduced by Bethe and formalized and extended to lower energies by Gribakin and Flambaum , (as in highly energetically favorable transition-state collision complexes); the satisfaction of the WKB criterion of Julienne and Mies, Gribakin and Flambaum, , Mody et al., and Côté et al. (as in many two-body ultracold molecular scattering processes, ranging from exothermic product formation to excitation exchange), and/or the absence of quantum reflection and quantum interference in ultracold inferno theory. Theoretical and experimental evidence in the past decade − ,,,− ,− supports the use of classical, semiclassical, and quasiclassical methods to simulate these systems, not only approximately, but nearly exactlyespecially important in cases where exact quantum mechanical approaches are computationally intractable today. This accuracy has the capacitywith careful considerationto enable simulation of aspects of a broad range of cold and ultracold molecular collisions within experimental accuracy that would be otherwise inaccessible.

Rationales for classical, semiclassical, and quasiclassical approaches to ultracold molecular collisions also provide a benchmark of where quantum effects are in fact essential. Quantum reflection and quantum interference determine the limitations of straightforward classical approaches. , However, both of these points are often correctable. Quantum reflection effects can be incorporated in terms of quantum suppression factors, , and quantum interference can be accounted for semiclassically via Van Vleck–Morette–Gutzwiller propagator/Green’s function, ,, Gutzwiller trace formula, , and wavepacket propagation methods. − Quantum resonances are also amenable to accurate semiclassical treatment, as exemplified by Roudnev and Cavagnero’s accurate semiclassical prediction of resonance positions in ultracold dipolar collisions in applied electric fields according to the WKB/Bohr-Sommerfeld quantization condition. , At times, classical calculations may be cumbersome. For example, the common assumption that collision partners may be represented as point particles may belie the importance of internal motion and its quantization, ,− and the assumption of ergodicity in RRKM-based calculation of collision complex lifetimes − may entail careful consideration of the true manifold on which classical trajectories are ergodic or alternative approaches to ameliorate nonergodicity in the considered range of phase space.

What is not addressed in this review and is not possible to simulate efficiently classically with modern methods is entanglement. This lacuna opens an opportunity for quantum computers of interest for the simulation of entanglement in chemistry; a topic that is especially important given leading-edge research into entanglement in ultracold molecules for reaction control, quantum sensing, and quantum computing, ,,− including in the presence of electromagnetic fields. ,

Theoretical prediction of quantum-to-classical crossover in the ultracold regime by a wide variety of authors (see refs − , , , , , , , , , , , − ) throws down the gauntlet for the wider experimental measurement of classical behaviors in the sub-mK regime. The novel picture of near-threshold cross sections given by recent classical and semiclassical approaches points to a key observable, namely the factor k scaling change in the cross section from the quantum threshold to the classical post-threshold law. A search is also underway for how to incorporate quantum effects and nonstatistical behavior into quasiclassical RRKM ,,,− and classical trajectory ,,, approaches in order to compute quantitative collision complex lifetimes. In particular, the hunt is on for a solution to unravel the mystery of why quasiclassical collision complex lifetimes accurate for certain ultracold collisions yield 5 orders of magnitude of discrepancy for others, with recent focus placed on examination of the role of conical intersections by Bause et al. and nuclear spin state coupling by Jachymski, Gronowski, and Tomza. Additionally, the longstanding predictions surrounding classical scattering cross sections for three-dimensional Coulomb potentials, multiple ionization, and atomic fragmentation invite a search for ultracold analogs beyond traditional atom–surface sticking systems in order to identify the extreme limits of quantum-classical correspondence to probe the nature of the interface between quantum and classical mechanics.

An additional theoretical area that remains to be further explored is how to address the severe impact that small variations in potential energy surfaces frequently have on scattering observables. Here, a 1% change in the potential (less than the uncertainty of contemporary ab initio methods) can exact a change of almost 10 orders of magnitude on the scattering cross section. − This means that even where classical, semiclassical, and quasiclassical approaches to cold and ultracold collisions are highly accurate, the result may still be wildly inaccurate. Since the days of Gribakin and Flambaum, ,− there has been significant work to illustrate this problem via a potential scaling technique ,,,,,− and to leverage this problem to probe potentials in terms of their quantum resonances − and Ramsauer-Townsend minima. Recently, statistical methods have been developed in an effort to ameliorate this problem. In 2015, Cui, Li, and Krems − instituted a Gaussian process model capable of producing the relationship between desired observables and potential and control parameters to high accuracy with few training points. For example, the Gaussian process model produced collision lifetimes of near-threshold C₆H₆ – Ar with a scaled root-mean-square error of just 5.09% for 40 classical-trajectory-derived training points. Likewise, in 2019 and 2020, Morita, Kłos, Krems, and Tscherbul , established a technique to capture observable variation by performing an ensemble of quantum scattering observables for potentials of varying strength to yield cumulative probability distributions for desired observables. These distributions exhibited universal behavior independent of the basis set and low-cost convergence with maximum quantum number N as low as N _max = 5 for Li – CaH collisions. These advancements point toward a continued role of data science and classical machine learning approaches to take full advantage of the limited knowledge of short-range potential energy surfaces and to peer into the mechanisms driving the cold and ultracold potential sensitivity.

The ongoing successes of classical, semiclassical, and quasiclassical approaches in the prototypically quantum regime encourage their resurgence at the standard chemistry occurring at higher temperatures. Chemistry stands to benefit at two levels: Computationally, given that the cost grows exponentially with the number of degrees of freedom quantum mechanically, classical approaches offer an exponential cost reduction. This reduction in cost suggests that the methods employed for ultracold molecules detailed here, as well as as quasiclassical trajectories and mixed/quantum classical dynamics − and additional semiclassical and path integral methods ,− may be further applied to bring accurate dynamics and scattering within reach for wide ranger of systems, thereby offering a tool complementary to current multiconfiguration time-dependent Hartree − and tensor-network methods. − Analytically, given that quantum approaches may necessitate visualization of molecular processes in terms of highly multidimensional wavepackets, classical approaches allow the envisionment of molecular processes in terms of readily interpretable particles, which facilitates elucidation of scattering and reaction mechanisms. This is especially relevant today given the current interest in simulation of molecules in the prototypical quantum regime to be the basis of molecular qubits − and quantum sensors. −

Finally, it is worth noting that the rationales put forward for classical, semiclassical, and quasiclassical approaches to ultracold molecular collisions make no assumption on the underlying particles in question. Thus, the same phenomenaclassical post-threshold laws, ,,,,− (quasi)­classical collision complex decay times, − etc.are expected to have a parallel life in near-threshold systems of other fields. Consider quantum threshold laws as a historical example. First developed in nuclear physics via the continuity approach of Bethe, Wigner R-matrix theory, and effective range theory; , the same quantum threshold laws have since been rederived in atomic physics via quantum defect theory , and multichannel quantum defect theory ,, and its advancements ,,,,− (a theory that is in fact mathematically equivalent to nuclear R-matrix theory , ). The classical post-threshold laws described here are therefore expected to follow in fields including positron and positronium collisions, ,,,− pion and hadron atom collisions, , plasma science, − and astrophysics and astrochemistry. − This universality of classical behavior close to threshold hints at a possible widespread impact of the ongoing molecular classical revival in today’s quantum era.

Biographies

Micheline B. Soley is an assistant professor in the Department of Chemistry at the University of Wisconsin–Madison, where she is affiliate of the Department of Physics and Data Science Institute as well as a member of the Wisconsin Quantum Institute and Chicago Quantum Exchange. Before her appointment as an assistant professor, she was a Yale Quantum Institute Postdoctoral Fellow at Yale University and a National Center for Supercomputing Applications Blue Waters Fellow, Harvard Merit/Graduate Society Term-time Research Fellow, and National Science Foundation Graduate Research Fellow at Harvard University. She was also a Fulbright Fellow at the Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy. In 2023, Micheline B. Soley was awarded the American Chemical Society Kavli Emerging Leader Award for her work to harness the power of data science in chemistry, based on her work on computing and chemistry deep in the quantum regime with an emphasis on ultracold collisions, quantum control, and parity-time reversal symmetry and the intersection of quantum computing and tensor-network methods.

Eric J. Heller is the Abbott and James Lawrence Professor of Chemistry and Professor of Physics at Harvard University, where he first joined as director of the Institute for Theoretical Atomic, Molecular, and Optical Physics (ITAMP) and Center for Astrophysics (CfA) in 1993. His current research on wave phenomena spans from chemistry to oceanography, with current research foci on time-dependent quantum mechanics, semiclassical physics, and classical-quantum correspondence in devices and molecular systems. In 2013, Eric J. Heller published Why You Hear What You Hear: An Experiential Approach to Sound, Music, and Psychoacoustics at Princeton University Press, the first textbook for nonspecialists on the physics of acoustics and psychoacoustics. In 2018, he followed this text with The Semiclassical Way to Dynamics and Spectroscopy to provide a holistic view into a semiclassical approach to quantum mechanics in molecular systems. For his discoveries, Eric J. Heller has been elected to numerous societies, including the American Academy of Arts and Sciences, the American Philosophical Society, and the National Academy of Science.

The authors declare no competing financial interest.