Two-body loss rates for reactive collisions of cold atoms

We present an effective two-channel model for reactive collisions of cold atoms. It augments elastic molecular channels with an irreversible, inelastic loss channel. Scattering is studied with the distorted-wave Born approximation and yields general expressions for angular momentum resolved cross sections as well as two-body loss rates. Explicit expressions are obtained for piecewise constant potentials. A pole expansion reveals simple universal shape functions for cross sections and two-body loss rates in agreement with the Wigner threshold laws. This is applied to collisions of metastable ${}^{20}\mathrm{Ne}$ and ${}^{21}\mathrm{Ne}$ atoms, which decay primarily through exothermic Penning or associative ionization processes. From a numerical solution of the multichannel Schrödinger equation using the best currently available molecular potentials, we have obtained synthetic scattering data. Using the two-body loss shape functions derived in this paper, we can match these scattering data very well.

Figure 1

Effective two-channel reaction model: molecular scattering channel 1 coupled to loss channel 2. Radial molecular potentials $v_{11}\left(r\right)$ (solid, blue line) and loss potential $v_{22}\left(r\right)$ (dashed, red line) are coupled by $v_{12}\left(r\right)$ (dashed-dotted, black line). At large separations, the potentials approach the channel thresholds ${\mathrm{\Delta }}_{ii}$ with ${\mathrm{\Delta }}_{11}>{\mathrm{\Delta }}_{22}$ and all couplings $v_{ij}\left(r\right)$ vanish. The straight thin dashed black lines indicate bound states.

Figure 2

Effective two-channel square-well potential vs radius $r$. Within the range $r<1$, the potentials $v_{11}$ (solid, blue line), $v_{22}$ (dashed, red line), and $v_{12}$ (dashed-dotted, black line) are constant. For radii $r>1$, the potential matrix decouples into the channel thresholds ${\mathrm{\Delta }}_{11}$ and ${\mathrm{\Delta }}_{22}$. The channel wave numbers $k_1, k_2$ are defined as in Eq. (3). The straight thin dashed black lines indicate bound states.

Figure 3

Zero contour lines of the $s$-wave Jost function $\Re {\mathcal{f}}_{0,11}^{\left(0\right)}\left(k_1\right)=0$ (solid line) and $\Im {\mathcal{f}}_{0,11}^{\left(0\right)}\left(k_1\right)=0$ (dashed line) in the complex $k_1$ plane for a potential depth ${\kappa }_1^2=4$. The black dot indicates the position of a bound state at $k_1=i{\mathcal{K}}_1=0.638i$ and corresponds to a binding energy of $E_r=-{\mathcal{K}}_1^2=-0.407$.

Figure 4

Exact scattering phase ${\eta }_l\left(k_1\right)$ (solid line) and single-pole approximation ${\eta }_l^p\left(k_1\right)$ (dashed line) versus $k_1$ for the $l=0$ (black, $\diamond$), $l=1$ (blue, $\square$), and $l=2$ (red, $\circ$) partial waves.

Figure 5

The exact partial cross section ${\sigma }_{l,11}^{\left(0\right)}\left(k_1\right)$ (solid line) and the single-pole approximation (dashed line) versus $k_1$ for $l=0$ (black, $\diamond$), $l=1$ (blue, $\square$), and $l=2$ (red, $\circ$) partial waves are almost indistinguishable. Using only the Breit-Wigner approximation ${\sigma }_{l=1,11}^{\left(0\right)p,r}\left(k_1\right)$ (dotted, blue line) of Eq. (58) leads to significant deviations in shape and resonance position $k_r=\sqrt{E_r}=0.530$ (vertical black line).

Figure 6

Zero contour lines of the $p$-wave Jost function $\Re {\mathcal{f}}_{1,11}^{\left(0\right)}\left(k_1\right)=0$ (solid line) and $\Im {\mathcal{f}}_{1,11}^{\left(0\right)}\left(k_1\right)=0$ (dashed line) in the complex $k_1$ plane for a potential depth ${\kappa }_1^2=9$. The black dots indicate the positions ${\mathcal{k}}_1=0.539-0.100i$ and $-{\mathcal{k}}_1^{*}$.

Figure 7

Zero contour lines of the $d$-wave Jost function $\Re {\mathcal{f}}_{2,11}^{\left(0\right)}\left(k_1\right)=0$ (solid line) and $\Im {\mathcal{f}}_{2,11}^{\left(0\right)}\left(k_1\right)=0$ (dashed line) in the complex $k_1$ plane for a potential depth ${\kappa }_1^2=19.5$. The black dots indicate the positions ${\mathcal{k}}_2=0.628-0.009i$ and $-{\mathcal{k}}_2^{*}$.

Figure 8

Zero contour lines of the $s$-wave Jost functions in channels 1 and 2: $\Re {\mathcal{f}}_{0,11}^{\left(0\right)}\left(k_1\right)=0$ (solid, black line), $\Im {\mathcal{f}}_{0,11}^{\left(0\right)}\left(k_1\right)=0$ (dashed, black line), and $\Re {\mathcal{f}}_{0,22}^{\left(0\right)}\left(k_1\right)=0$ (solid, gray line), $\Im {\mathcal{f}}_{0,22}^{\left(0\right)}\left(k_1\right)=0$ (dashed, gray line) in the complex $k_1$ plane. The black dot indicates the position $k_1=i{\mathcal{K}}_1=0.638i$. The potential parameters are given in Table 2.

Figure 9

Zero contour lines of $p$-wave Jost function in channels 1 and 2: $\Re {\mathcal{f}}_{1,11}^{\left(0\right)}=0$ (solid, black line), $\Im {\mathcal{f}}_{1,11}^{\left(0\right)}=0$ (dashed, black line), and $\Re {\mathcal{f}}_{1,22}^{\left(0\right)}=0$ (solid, gray line), $\Im {\mathcal{f}}_{1,22}^{\left(0\right)}=0$ (dashed, gray line) in the complex $k_1$ plane. The black dots indicate the positions ${\mathcal{k}}_1=0.539-0.100i$ and $-{\mathcal{k}}_1^{*}$. The potential parameters are given in Table 2.

Figure 10

DWBA partial-wave two-body loss rates ${\beta }_l^{\left(1\right)}$ (solid line) and the single-pole approximation ${\beta }_l^{\left(1\right)p}$ (dashed line) versus channel wave number $k_1$ for $l=0$ (black, $\diamond$), $l=1$ (blue, $\square$), $l=2$ (red, $\circ$) for the potential parameters of Table 2.

Figure 11

Synthetic scattering two-body loss rates ${\beta }^{\text{synt}}$ (solid, green, $△$) and pole expansion ${\beta }^p$ (dashed, green, $△$) with the coefficients of Table 3 versus relative collisions energy $E_{\text{rel}}=\hslash k^2/2\mu$ in units of $k_B$ for ${}^{21}\mathrm{Ne}\text{and}{}^{20}\mathrm{Ne}$ collisions. Additionally shown are the partial-wave contributions $1/2\times {\beta }_l^{\text{synt}}$ [solid, (black, $\diamond$), (blue, $\square$), (red, $\circ$)], $1/2\times {\beta }_l^p$ [dashed, (black, $\diamond$), (blue, $\square$), (red, $\circ$)] for $l=0,1,2$, respectively, and experimental data point (green filled circle with error bars).