Universality and chaoticity in ultracold K+KRb chemical reactions

Abstract

A fundamental question in the study of chemical reactions is how reactions proceed at a collision energy close to absolute zero. This question is no longer hypothetical: quantum degenerate gases of atoms and molecules can now be created at temperatures lower than a few tens of nanokelvin. Here we consider the benchmark ultracold reaction between, the most-celebrated ultracold molecule, KRb and K. We map out an accurate ab initio ground-state potential energy surface of the K₂Rb complex in full dimensionality and report numerically-exact quantum-mechanical reaction dynamics. The distribution of rotationally resolved rates is shown to be Poissonian. An analysis of the hyperspherical adiabatic potential curves explains this statistical character revealing a chaotic distribution for the short-range collision complex that plays a key role in governing the reaction outcome.

Figure 1. K–KRb potential energy surface.

(a) A two-dimensional cut through the energetically lowest ²A′ (blue curve) and ²B′ (red curve) adiabatic potential energy surfaces of the KRbK trimer as a function of the K−Rb bond lengths and the angle between K−Rb−K along the isosceles C_2ν geometry with R_K1Rb=R_K2Rb. The base of the figure shows the corresponding contour graph. The zero of energy corresponds to the energy of three well-separated atoms. The unit of length is the Bohr radius a₀=0.0529177, nm and that of energy is wavenumbers. (b) A contour graph of the K−Rb−K potentials based on the data in a. Contour labels are in units of 10³ cm⁻¹. In both panels the seam of conical intersections is shown by the green line.

Figure 2. Energetics and reaction rates.

(a) Energetics of the KRb+K→K₂+Rb reaction. The j=0 vibrational levels of the KRb X¹Σ⁺ potential and j′=0 vibrational levels of the K₂ potential are shown by black and red lines, respectively. The grey shaded area indicates the closely spaced energetically allowed rotational levels of K₂. The zero of energy is located at the dissociation limit of KRb and K₂. (b) Reaction rate coefficients from J=0 EQM calculations based on either the full (black curves) or pairwise (red curves) potential as a function of collision energy in units of the Boltzmann constant. The total and vibrationally resolved reaction rate coefficients are shown. The green curve is the s-wave unitary rate coefficient for atom-dimer scattering. The closed circle with error bars (one s.d.) corresponds to an experimental measurement taken at a temperature of 250 nK. (c) Total reaction rate coefficient (green curve) as a function of temperature based on the J-shifting method and the J=0 EQM results for the full trimer potential. The black curve repeats this latter curve from (b) as a function of energy. The red curve shows the rate coefficient of the s-wave universal model (UM) as a function of collision energy, while the blue curve shows the UM data including all relevant partial waves as a function of temperature.

Figure 3. Rotationally resolved rates.

The state-to-state J=0 EQM reaction rate coefficients from the v=0, j=0 ro-vibrational state of KRb to the v′=0, 1 and 2 vibrational level of K₂ as a function of product rotational quantum number j′. The blue and red bars are from calculations with the full trimer and additive pairwise potential, respectively. Rates are for an initial collision energy of E/k=210 nK.

Figure 4. Probability distribution of rotationally resolved rates.

Distribution of the j′-resolved rate coefficient. Blue and red markers correspond to J=0 EQM data populating the v′=0, 1 and 2 of K₂ at initial collision energy E/k=210 nK and 1 K, respectively. Results for full and pairwise trimer potentials are shown by different markers. For each collision energy the rate coefficients are scaled to its mean value. The solid black curve is the Poisson distribution.

Figure 5. Statistical analysis of the short-range adiabatic potentials.

(a) Distribution of nearest-neighbor spacings of J=0 KRbK and LiYbLi adiabatic potential energies in hyperspherical coordinates. Each black, dotted curve corresponds to the distribution for a single hyper radius ρ as a function of scaled level spacing. The shaded grey area and red curve are Wigner–Dyson and Poisson distributions, respectively. For both KRbK and LiYbLi we use the full trimer potential. (b) The Brody parameter q as a function of hyper radius for both KRbK and LiYbLi as derived from the data in (a) for the full trimer potential and data for KRbK based on the pair-wise potential. Blue and red markers correspond to distributions for KRbK obtained with the full and pairwise electronic potentials, respectively, whereas the orange markers for LiYbLi are obtained with the full potential. For q→0 the distribution approaches the Poisson distribution while for q→1 it approaches the Wigner–Dyson distribution.

Figure 6. Dimer and trimer potential energy curves.

The calculated singlet X¹Σ⁺ and triplet a³Σ⁺ potentials of KRb (solid lines in a) and K₂ (solid lines in b) as a function of inter-atomic separation using the same basis set and core polarization potential as for the KRbK trimer. The dashed lines in both panels are the corresponding spectroscopically accurate dimer potentials. (c) A one-dimensional cut through the two energetically lowest adiabatic KRbK trimer potential surfaces based on full (solid lines) and pairwise (dashed lines) calculations for R_K(1)Rb=R_K(2)Rb=10a₀.

Figure 7. K–KRb potential energy surface for collinear geometry.

A two-dimensional cut through the energetically lowest ²A′ adiabatic potential energy surface of the KRbK trimer as a function of the K−Rb and K−K bond lengths along the collinear geometry. The zero of energy corresponds to the energy of three well-separated atoms. The arrows indicate the entrance and exit channels of the chemical reaction.