Figure 1

Optical configuration for the 3DSOC. (a) Four states are coupled using two-photon optical transitions. (b) The momentum-space displacement vectors ${\mathbf{K}}_i$ point to the four vertices of a tetrahedron. (c) The fluxes through the surface are chosen to be ${\Phi }_i=\pi /2$ modulo $2\pi$.Reuse & Permissions

Figure 2

Four hyperfine states $|F,m_F⟩$ of ${}^{87}\mathrm{Rb}$ are coupled using nine lasers. The quantization axis is set by a Zeeman field along the $\widehat{z}$ axis. The couplings are produced in pairs. (a) The four states in the tetrahedral coupling are mapped to physical states according to $|1⟩=|2,0⟩$, $|2⟩=|1,+1⟩$, $|3⟩=|1,0⟩$, $|1⟩=|2,+1⟩$. (b) The frequencies of the three sets of lasers are given by $\{{\omega }_a,{\omega }_a+{\delta }_{12},{\omega }_a+{\delta }_{34}\}$ (dashed blue), $\{{\omega }_b,{\omega }_b+{\delta }_{13},{\omega }_b+{\delta }_{24}\}$ (dotted black), and $\{{\omega }_c,{\omega }_c+{\delta }_{14},{\omega }_c+{\delta }_{23}\}$ (solid red), where ${\delta }_{ij}={\omega }_i-{\omega }_j$ is the frequency difference between the states $|i⟩$ and $|j⟩$ in the rotating frame. (c) The geometry of the nine laser beams. $-{\mathbf{k}}_{12}=-{\mathbf{k}}_{13}={\mathbf{k}}_{23}={\mathbf{k}}_{34}=k_L{\widehat{e}}_{-}$, ${\mathbf{k}}_a={\mathbf{k}}_{23}=-{\mathbf{k}}_{14}=k_L\widehat{z}$ and ${\mathbf{k}}_b=-{\mathbf{k}}_c=k_L{\widehat{e}}_{-}$. The unit vectors ${\widehat{e}}_{\pm }=\pm \frac{1}{\sqrt{2}}(\widehat{x}\pm \widehat{y})$. For a complete description of the laser parameters, see the Supplemental Material [21].Reuse & Permissions

Figure 3

Energy spectrum of the isotropic 3DSOC. (a) Energy spectrum of spherical ground state manifold $|\mathbf{p}|=mv$ with $E=-m{v}^2/2$. Near the ground state manifold the dispersion $E(\delta p)$ is approximately parabolic in a small deviation $\delta p$ from the momentum of the ground state. (b) Shells of constant energy, $E_n=-(1-0.1n)m{v}^2/2$ with $n=0$, 1, 2, 3. These low energy shells have smaller surface area until $\mathbf{p}=0$. (c) A one dimensional cut of the energy spectrum. The full spectrum is generated by rotating this spectrum along three axes. The blue (red) band corresponds to states with momentum aligned (antialigned) with spin.Reuse & Permissions