Ultracold collisions in the presence of synthetic spin-orbit coupling

We present an analytical description for ultracold collisions between two spin-$\frac{1}{2}$ fermions with isotropic spin-orbit coupling (SOC) of the Rashba type. We show that regardless of how weak the SOC may be, at sufficiently low energies the collision properties are significantly modified. The ubiquitous low energy Wigner threshold behavior is changed and its modified form for ground-state neutral atoms is given. The particles are further found to scatter preferentially into the lower-energy helicity states due to the breaking of parity conservation. This latter point establishes interaction with SOC as one mechanism for the spontaneous emergence of handedness. Our theory is applicable both to elementary spin-$\frac{1}{2}$ fermions such as electrons in condensed matter and to ultracold pseudo-spin-$\frac{1}{2}$ atoms such as ${}^6$Li in its ground hyperfine state.

Figure 1

The dispersion curve for the helicity state $|\pm ,{\mathit{n}}_{\mathrm{so}}\rangle |\mathit{k}\rangle$ of a single pseudo-spin-$\frac{1}{2}$ atom with isotropic Rashba SOC. The two branches correspond, respectively, to the higher energy (“$+$”) and lower energy (“$-$”) states when $C_{\mathrm{so}}>0$. At each energy $E={\hslash }^2{k}^2/2m>0$, the “$\pm$” states correspond to different $k_{+}$ and $k_{-}$. When $E<0$, the two intersection points are labeled by $k_{-}^{<}$ and $k_{-}^{>}$, respectively. Both belong to the same lower energy helicity branch. $s_E={\hslash }^2{k}_{\mathrm{so}}^2/2\mu$ is the energy scale for the SOC.

Figure 2

The three branches of dispersion for two particles with SOC in the center-of-mass frame. For each energy $E={\hslash }^2{k}^2/2\mu >0$, the three corresponding $k$'s, given in the order of increasing magnitude, are $k_1=\sqrt{k_{\mathrm{so}}^2+k^2}-k_{\mathrm{so}}$ for the $|+,+;{\mathit{k}}_1\rangle$ state, $k_2=k$ for the $|+,-;{\mathit{k}}_2\rangle$ and $|-,+;{\mathit{k}}_2\rangle$ states, and $k_3=\sqrt{k_{\mathrm{so}}^2+k^2}+k_{\mathrm{so}}$ for the $|-,-;{\mathit{k}}_3\rangle$ state. As in Fig. 1 for a single atom, for $E<0$ the two intersection points are labeled by $k_3^{<}$ and $k_3^{>}$, respectively, belong to the same lowest energy branch.

Figure 3

The $k$ dependence of the cross sections $\sigma {\left[\right|1\rangle }_{\mathrm{in}}\to {|1\rangle }_{\mathrm{out}}]$ and $\sigma {\left[\right|3\rangle }_{\mathrm{in}}\to {|1\rangle }_{\mathrm{out}}]$ of Eq. (26). The limits of small $k$ are discussed in the main text. The dotted line is the $\sim 1/k^2$ unitarity limit, while the dot-dashed lines denote the modified Wigner threshold limit of $\sim k^4$ at small $k$.

Figure 4

The same as in Fig. 3 but for the scattering cross sections $\sigma {\left[\right|1\rangle }_{\mathrm{in}}\to {|3\rangle }_{\mathrm{out}}]$ and $\sigma {\left[\right|3\rangle }_{\mathrm{in}}\to {|3\rangle }_{\mathrm{out}}]$ of Eq. (27). The modified threshold limit becomes $\sim a_{F=0}^2$ and $\sim 1/k_{\mathrm{so}}^2$, respectively, for $k_{\mathrm{so}}{a}_{F=0}\ll 1$ and $k_{\mathrm{so}}{a}_{F=0}\gg 1$.

Figure 5

The universal ratios of inelastic scattering cross sections, $\sigma {\left[\right|1\rangle }_{\mathrm{in}}\to {|3\rangle }_{\mathrm{out}}{]/\sigma \left[\right|3\rangle }_{\mathrm{in}}\to {|1\rangle }_{\mathrm{out}}]$, with (solid line) and without SOC (dash-dot line), as a function of $k$. The result with SOC is guaranteed by the time-reversal symmetry to be valid at all energies. The result without SOC is guaranteed by a combination of time-reversal and parity conservations to be valid at all energies. The difference is due to the break of parity conservation by SOC.