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

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.