Strong dependence of ultracold chemical rates on electric dipole moments

We use the quantum threshold laws combined with a classical capture model to provide an analytical estimate of the chemical quenching cross sections and rate coefficients of two colliding particles at ultralow temperatures. We apply this quantum threshold model (QT model) to indistinguishable fermionic polar molecules in an electric field. At ultracold temperatures and in weak electric fields, the cross sections and rate coefficients depend only weakly on the electric dipole moment $d$ induced by the electric field. In stronger electric fields, the quenching processes scale as $d^{4(L+\frac{1}{2})}$ where $L>0$ is the orbital angular-momentum quantum number between the two colliding particles. For $p$-wave collisions ($L=1$) of indistinguishable fermionic polar molecules at ultracold temperatures, the quenching rate thus scales as $d^6$. We also apply this model to pure two-dimensional collisions and find that chemical rates vanish as $d^{-4}$ for ultracold indistinguishable fermions. This model provides a quick and intuitive way to estimate chemical rate coefficients of reactions occuring with high probability.

Figure 1

Effective potential barrier $V(R)$ as a function of the intermolecular separation $R$. $V_b$ and $R_b$ denote the height and the position of the centrifugal barrier.

Figure 2

Quenching cross section of ${}^{39}\mathrm{K}+{}^{39}{\mathrm{K}}_2$ as a function of the collision energy for the partial wave $L=1-5$: (i) calculated with a full quantum calculation (solid lines), reproduced from Ref. [10], (ii) using the QT model (dashed lines), (iii) fitting the QT model (dotted line), using $p=0.34$ in Eq. (14).

Figure 3

Quenching probability of ${}^{39}\mathrm{K}+{}^{39}{\mathrm{K}}_2$ as a function of the collision energy for the partial wave $L=1$: comparison between the full quantum calculation (solid line) and the QT model (dashed line). The fitted QT model appears as a dotted line. The height of the barrier $V_b$ and the corrected height $\gamma \times V_b(\gamma =2.06)$ appear as vertical lines.

Figure 4

Diabatic (dashed lines) and adiabatic (solid lines) barrier heights $V_b^{d,a}$ as a function of the induced dipole moment $d$ for the partial waves $L=1,M_L=0$ (red, lower) and $L=1,\left|M_L\right|=1$ (blue, upper).

Figure 5

Quenching rate coefficients of two indistinguishable fermionic polar ${}^{40}\mathrm{K}$${}^{87}\mathrm{Rb}$ molecules as a function of the induced electric dipole moment for $L=1$ and for a temperature of $T=350$ nK (black curves). The rates have been calculated using the barrier heights of Fig. 4. The red lines represent the $L=1,M_L=0$ partial-wave contribution. The blue lines represent the sum of $L=1,M_L=1$ and $L=1,M_L=-1$ partial-wave contributions. The dashed lines represent the rates calculated with the diabatic barriers while the solid lines with the adiabatic barriers (see text for detail). The total, $M_L=0$ and $\left|M_L\right|=1$ curves have been indicated in the left-hand side.

Figure 6

Same as Fig. 5 but we use the analytical expressions for the rates (see text for detail). The total, $M_L=0$ and $\left|M_L\right|=1$ curves have been indicated in the left-hand side. The individual analytical curves have been indicated in the right-hand side by roman numbers.

Figure 7

Barrier heights as a function of the induced dipole moment $d$ for the partial waves $\left|M\right|=1$ in two dimensions. The green thin curves represent the analytical Eq. (28) (constant) and Eq. (29) ($d^4$). The dashed black curve is the sum of them. The solid black curve is the height of the barrier in Eq. (24).

Figure 8

Two dimensional quenching rate coefficient (black curves) of two indistinguishable fermionic polar ${}^{40}{\mathrm{K}}^{87}$Rb molecules as a function of the induced electric dipole moment for the $M=1$ and $M=-1$ components at a temperature of $T=350$ nK. The dashed lines represent the rate using analytical expressions while the solid line represents the rate using the numerical expression (see text for detail). The individual analytical curves have been indicated in the right-hand side by roman numbers.