Reactive collisions of two ultracold particles in a harmonic trap

We consider two ultracold particles confined in a spherically symmetric harmonic trap and interacting via isotropic potential with absorbing boundary conditions at short range that models reactive scattering. First, we apply the contact pseudopotential with complex scattering length and investigate the properties of eigenenergies and eigenfunctions for different trap states as a function of the real and imaginary parts of the scattering length. In the analyzed case, the eigenenergies take complex values and their imaginary part can be interpreted in terms of decay rates. Then we introduce the model of a square well with absorbing boundary conditions that is later used to investigate the effects of the finite range of interaction on the properties of a reactive scattering in the trap. Finally, we analyze the decay rates for some reactive alkali dimer molecules assuming short-range probability of reaction equal to unity. In this case, the complex scattering length is given by the universal value related to the mean scattering length of the van der Waals interaction between molecules.

Figure 1

Real (upper panel) and imaginary (lower panel) parts of the complex scattering length for van der Waals potential and for different values of $s$ and $y$ parameters describing the short-range behavior of the wave function.

Figure 2

$\text{Re}E$ (left panel) and $\text{Im}E$ (right panel) dependence on $\text{Re}a$ for various $\text{Im}a$ values.

Figure 3

$\text{Re}E$ (left panel), $\text{Im}E$ (central panel) dependence on $\text{Re}a$ for various $\text{Im}a$ values, and corresponding radial wave functions (right panel) in the vicinity of an avoided crossing between $|E_0|$ and $|E_1|$.

Figure 4

Absorbing square-well potential.

Figure 5

$\text{Re}{\tilde{a}}_{\eta }$ (top) and $\text{Im}{\tilde{a}}_{\eta }$ (bottom) dependence on $\alpha$ for various $\eta$ phases (left panel) and for various $\eta$ moduli (right panel).

Figure 6

Finite-range interaction potential between molecules in a harmonic trap.

Figure 7

$\text{Re}E$ (left panel) and $\text{Im}E$ (right panel) dependence on $u$ for various $\eta$ values. Black dots: the zero-range interaction model with energy-independent scattering length; green dots: the zero-range interaction model with energy-dependent scattering length; blue empty diamonds: exact solution for the absorbing square-well model. The square-well diameter $L=0.2d$.

Figure 8

Lifetime dependence on harmonic trap frequency for two colliding ${}^{41}\mathrm{K}{}^{87}\mathrm{Rb}$ (black curves) and ${}^7\mathrm{Li}{}^{133}\mathrm{Cs}$ (green curves) molecules in the universal limit. Solid, dotted, and circle lines correspond, respectively, to the first ($E_1$), second ($E_2$), and third ($E_3$) excited states of the system of two molecules in a trap.