Formation of Feshbach molecules in the presence of artificial spin-orbit coupling and Zeeman fields

We derive general conditions for the emergence of singlet Feshbach molecules in the presence of artificial Zeeman fields for arbitrary mixtures of Rashba and Dresselhaus spin-orbit coupling in two or three dimensions. We focus on the formation of two-particle bound states resulting from interactions between ultracold $\text{spin-}1/2$ fermions, under the assumption that interactions are short ranged and occur only in the $s\text{-wave}$ channel. In this case, we calculate explicitly binding energies of Feshbach molecules and bound-state energy thresholds and analyze their dependence on spin-orbit couplings, Zeeman fields, interactions, and center-of-mass momentum, paying particular attention to the experimentally relevant case of spin-orbit couplings with equal Rashba and Dresselhaus (ERD) amplitudes.

Figure 1

Plots of the generalized two-particle helicity bands $E_{\Uparrow \Uparrow }(\mathbf{k},\mathbf{K})$ (black solid line), $E_{\Uparrow \Downarrow }(\mathbf{k},\mathbf{K})$ (red dot-dashed line), $E_{\Downarrow \Uparrow }(\mathbf{k},\mathbf{K})$ (green dashed line), and $E_{\Downarrow \Downarrow }(\mathbf{k},\mathbf{K})$ (blue dotted line) along the direction of relative momentum $\mathbf{k}=(k_x,0,0)$, for ERD spin-orbit coupling $v=k_R/m$, various values of detuning $h_y$ and Raman intensity $h_z$, and specific values of the center-of-mass momentum $\mathbf{K}=(K_x,0,0)$. The parameters used are (a) $h_y=0, h_z=0.5E_R$, and $K_x=0$; (b) $h_y=0.5E_R, h_z=1.0E_R$, and $K_x=0$; (c) $h_y=1.25E_R, h_z=0.5E_R$, and $K_x=0$; and (d) $h_y=0.5E_R, h_z=0.5E_R$, and $K_x=1.25k_R$. Notice the change in location of the minimum of $E_{\Uparrow \Uparrow }$ from finite $k_x$ in panels (a) and (b) to $k_x=0$ in panels (c) and (d).

Figure 2

Plots of bound-state energy $E_B/E_R$ vs $1/\left(k_R{a}_s\right)$ with $\mathbf{K}=\mathbf{0}, h_z=0, h_y=0$ are shown in panel (a) for $v=0$ (blue dotted line), $v=0.5k_R/m$ (green dashed line), $v=0.75k_R/m$ (red dot-dashed line), and $v=k_R/m$ (black solid line). Plots of $E_{\mathrm{Bin}}/E_R$ vs $1/\left(k_R{a}_s\right)$ with $\mathbf{K}=\mathbf{0}, v=k_R/m, h_y=0$ are shown in panel (b) for $h_z=0$ (blue dotted line), $h_z=E_R$ (green dashed line), $h_z=2E_R$ (red dot-dashed line), and $h_z=3E_R$ (black solid line). Plots of $E_{B,\mathrm{th}}/E_R$ and $1/\left(k_R{a}_{s,\mathrm{th}}\right)$ vs $h_z/E_R$ with $\mathbf{K}=\mathbf{0}, h_y=0$ are shown respectively in panels (c) and (d) for $v=0.25k_R/m$ (blue dotted line), $v=0.5k_R/m$ (green dashed line), $v=0.75k_R/m$ (red dot-dashed line), and $v=k_R/m$ (black solid line).

Figure 3

(a) Plots of $E_B/E_R$ vs $1/\left(k_R{a}_s\right)$ with $\mathbf{K}=\mathbf{0}, h_y=0.25E_R$, and $h_z=0.5E_R$ for $v=0.05k_R/m$ (blue dotted line), $v=0.25k_R/m$ (green dashed line), $v=0.75k_R/m$ (red dot-dashed line), and $v=k_R/m$ (black solid line). (b) Plots of $E_{\mathrm{Bin}}/E_R$ vs $1/\left(k_R{a}_s\right)$ with $\mathbf{K}=\mathbf{0}, h_z=E_R$, and $v=k_R/m$ for $h_y=0$ (blue dotted line), $h_y=E_R$ (green dashed line), $h_y=2E_R$ (red dot-dashed line), and $h_y=3E_R$ (black solid line). Plots of $E_{B,\mathrm{th}}/E_R$ and $1/\left(k_R{a}_{s,\mathrm{th}}\right)$ vs $h_y/E_R$ with $\mathbf{K}=\mathbf{0}$, and $v=k_R/m$ are shown respectively in panels (c) and (d) for $h_z=0$ (black solid line), $h_z=E_R$ (red dot-dashed line), $h_z=2E_R$ (green dashed line), and $h_z=3E_R$ (blue dotted line).

Figure 4

Plots of bound-state threshold energies (solid black) and of energies of Feshbach molecules (blue dotted line with $1/\left(k_R{a}_s\right)=0.75$; green dashed line with $1/\left(k_R{a}_s\right)=1.25$; red dot-dashed line with $1/\left(k_R{a}_s\right)=1.75$) vs center-of-mass momentum $\mathbf{K}=(K_x,K_y=0,K_z=0)$ for $v=k_R/m$ and $h_z=0.5E_R$, with (a) $h_y=0$, (b) $h_y=0.5E_R$, (c) $h_y=1.5E_R$, and (c) $h_y=2.5E_R$. Notice the absence of inversion symmetry (parity) when $h_y\ne 0$.