Triatomic Photoassociation in an Ultracold Atom-Molecule Collision

TL;DR

This work addresses triatomic photoassociation in ultracold atom–molecule collisions by constructing a long-range, spin–orbit–inclusive theoretical framework for atom–dimer interactions. It uses a multipole expansion to derive a long-range Hamiltonian with leading terms $V(R) abla o C_6/R^6$ and $C_8/R^8$, and builds a product basis combining atomic channels $igl|nlj mjigr angle$ with dimer states in Hund’s case C, while transforming between frames to evaluate matrix elements. Short-range physics is modeled via boundary conditions at the van der Waals length, allowing calculation of continuum initial states and weakly bound final vibrational states; PA rates are computed from $S_ u(E)=igra ext{initial}ig|oldsymbol{d}oldsymbol{ ho}ig| ext{final}_ uigra$ and the rate expression $R_{PA}". The results show that PA is strongest when the dimer rotational state is the same (e.g., $N=0$) and is more pronounced for polar dimers (Cs–NaCs) than for nonpolar dimers (Cs$_2$), with consistency verified against Cs$_3$ PA data. The approach clarifies how long-range couplings govern ultracold triatomic PA and informs experimental prospects for controlled molecule formation in overlapped optical tweezers.

Abstract

Ultracold collisions of neutral atoms and molecules have been of great interest since experimental advances enabled the cooling and trapping of such species. This study is a theoretical investigation of a low-energy collision between an alkali atom and a diatomic molecule, accompanied by absorption of a photon from an external electromagnetic field. The long-range interaction between the two species is treated, including the atomic spin-orbit interaction. The long-range potential energy curves for the triatomic complex are calculated in realistic detail, while the short-range behavior is mimicked by applying different boundary conditions at the van der Waals length. The photoassociation (PA) rate of an atom colliding with a dimer is calculated for different alkali atoms, namely Na and Cs. The model developed in this study is also tested against known results for the formation rate of the Cs$_3$ complex via PA, namely to compare with the work done by Rios et al., PRL 115, 073201 (2015), and the results are in generally good agreement.

Figures (15)

• Figure 1: The blue coordinate system $\{X_D,Y_D,Z_D\}$ is chosen such that the dimer lies on the $Z_D$ axis. The black coordinate system $\{X_T,Y_T,Z_T\}$ is chosen such that the axis connecting the center of mass of the atom and the dimer is the $Z_T$ axis. The two coordinates are related by a proper Euler rotation $R(0,\beta,0)$ around their mutual y axis with the angle $\beta$. In this convention, the y axis for both coordinates points out of the page.
• Figure 2: The potential energy curves for the Cs-NaCs system are plotted as functions of the atom-dimer distance $R$. Each family of curves is associated with a single atomic channel, namely $\lvert6S;|\omega| =1/2\rangle$ (a), and $\lvert6P_{3/2};|\omega| = 1/2\rangle$ (b), and $\lvert6P_{3/2};|\omega| = 3/2\rangle$(c). Each dissociative channel corresponds to a different dimer rotational quantum number $N$, in the range $N = 0-7$. The zero of the energy scale is fixed at the ground state energy for the independent atom-dimer system.
• Figure 3: The potential energy curves for the Na-NaCs system are plotted as functions of the atom-dimer distance $R$. Each family of curves is associated with a single atomic channel, namely $\lvert3S;|\omega| =1/2\rangle$ (a), and $\lvert3P_{3/2};|\omega| = 1/2\rangle$ (b), and $\lvert3P_{3/2};|\omega| = 3/2\rangle$(c). Each dissociative channel corresponds to a different dimer rotational quantum number $N$, in the range $N = 0-7$. The zero of the energy scale is fixed at the ground state energy for the independent atom-dimer system.
• Figure 4: (a) The spectrum and the wave functions of the final states $\lvert3P_{3/2};N = 0;\epsilon_{\nu}<0;|\omega| = 1/2\rangle$ are plotted vs the internuclear distance $R$. Two different boundary conditions considered, namely $a = 223$, and $a = 78$. the inset shows the energy normalized radial wave functions for the initial state of Na-NaCs, $\lvert3S;N = 0;E\rangle$. The blue curve is the state with infinite scattering length while the red curve is the state with zero scattering length. (b) The normalized PA rate $K_{PA}$ is plotted for each final state at average collision energy $T = 200 nK$. Each color corresponds to different values for the scattering lengths as shown in the inset.
• Figure 5: (a) The spectrum and the wave functions of the final states $\lvert6P_{3/2};N = 0;\epsilon_{\nu}<0;|\omega| = 1/2\rangle$ are plotted vs the internuclear distance $R$. Two different boundary conditions considered, namely $a = 162$, and $a = 53$. the inset shows the energy normalized radial wave functions for the initial state of Cs-NaCs, $\lvert6S;N = 0;E\rangle$. The blue curve is the state with infinite scattering length while the red curve is the state with zero scattering length. (b) The normalized PA rate $K_{PA}$ is plotted for each final state at average collision energy $T = 200 nK$. Each color corresponds to different values for the scattering lengths as shown in the inset.