A rigorous adiabatic approach to ultracold atom-molecule collisions in a magnetic field

TL;DR

This work develops a rigorous adiabatic coupled-channel framework for ultracold atom–molecule collisions in external magnetic fields, combining an ACC formalism with a diabatic-by-sector propagation scheme. By solving the adiabatic eigenproblem ${\hat H}_{\text{ad}}\Phi_i = \epsilon_i(R)\Phi_i$ and expanding the total wavefunction in adiabatic basis states, the authors show that cross sections for Mg + NH collisions can be obtained with ~2× fewer channels than standard diabatic CC calculations. They introduce a log-derivative–based RBT protocol that dynamically reduces the propagation basis size across $R$, yielding 15–30× computational gains for the propagation step, especially at long range, and are capable of handling strong anisotropy and resonances. The approach remains accurate when a modest error in cross sections is tolerable, and it opens avenues for efficient modeling of magnetic Feshbach resonances and complex ultracold chemistry, with potential extensions to reactive scattering and additional molecular degrees of freedom.

Abstract

We extend the rigorous adiabatic coupled-channel formalism to ultracold nonreactive atom-molecule collisions in the presence of an external magnetic field. The wavefunction of the collision complex is expanded in adiabatic basis states obtained by solving the eigenvalue problem for the adiabatic Hamiltonian (the total Hamiltonian of the collision complex minus the radial kinetic energy) on a grid of atom-molecule distances $R$. The resulting coupled-channel equations are solved using the diabatic-by-sector method. We show that the adiabatic approach provides accurate cross sections for cold and ultracold Mg ($^1$S) + NH ($^3Σ^-$) collisions in a magnetic field with ~2 times fewer channels than the standard diabatic basis. We further develop an efficient $R$-dependent basis truncation protocol (RBT), in which the elements of the log-derivative matrix are sampled and discarded as it is propagated from small to large $R$. While RBT can be applied in both the adiabatic and diabatic bases, we show that the adiabatic basis can be reduced to just the open channels at long range, leading to an overall computational gain of ~15-30 for the propagation part of the calculation. The gain is particularly significant in situations where substantial errors in the calculated cross sections ($<$50\%) can be tolerated or long-range interactions are involved, making the adiabatic basis formulation a promising approach to strongly anisotropic collisions and chemical reactions in the presence of an external magnetic field.

Figures (16)

• Figure 1: Schematic of the diabatic-by-sector procedure for propagating the log-derivative matrix in the adiabatic basis. The $n^{\text{th}}$ sector is defined on an interval $R \in [a_n,b_n]$ by the starting point $R=a_n$, the midpoint $R=c_n$, and the endpoint $R=b_n$. The endpoint of the $n^{\text{th}}$ sector is the starting point of the following sector ($a_n = b_{n-1}$). The blue arrows show the propagation and sector-to-sector transformation of the log-derivative matrix.
• Figure 2: Schematic representation of the log-derivative matrix (${\bf Y}$) in the adiabatic basis as a square matrix of dimension $M_n \times M_n$. The log-derivative matrix is divided into four submatrices whose dimensions are based on the number of locally open channels ($M^{\text{(LO)}}_n$) and the number of locally closed channels ($M^{\text{(LC)}}_n$). The matrix is symmetric, so the green sections are transposes of one another.
• Figure 3: The $N=0$ (a), $N=1$ (b), and $N=2$ (c) energy levels of NH ($^3 \Sigma ^-$) plotted against the magnitude of the applied magnetic field. The initial state for scattering calculations is shown by the dashed line.
• Figure 4: (a) Lowest adiabatic potentials of the Mg-NH collision complex. The inset zooms into the minimum of the six deepest adiabatic potentials to highlight the Zeeman splitting. (b) Medium- to long-range adiabatic potentials of the Mg-NH collision complex. All potentials have a well-defined value of $M_{\text{tot}} = 1$ and are calculated at a magnetic field of $B = 1000$ G. The zero of energy corresponds to the asymptotic value of the Mg-NH interaction potential ($R \rightarrow \infty$). (c) Legendre expansion coefficients of the Mg-NH interaction potential.
• Figure 5: Elastic (dashed lines) and total inelastic (solid lines) cross sections as a function of collision energy computed for ultracold Mg + NH collisions calculated via the adiabatic formulation using $R$-dependent truncation with a threshold of $0.001 \text{ }a_0^{-1}$. Open shapes are cross sections calculated using the standard diabatic treatment with the uncoupled SF representation at magnetic field strengths 10 G (brown diamonds), 100 G (navy circles), 1000 G (teal squares). The upper panel (a) shows cross sections near a resonance at $B=100$ G, and the lower panel (b) shows cross sections at ultracold energies.