Spin-Orbit Coupling and Spin Textures in Optical Superlattices

We propose and demonstrate a new approach for realizing spin-orbit coupling with ultracold atoms. We use orbital levels in a double-well potential as pseudospin states. Two-photon Raman transitions between left and right wells induce spin-orbit coupling. This scheme does not require near resonant light, features adjustable interactions by shaping the double-well potential, and does not depend on special properties of the atoms. A pseudospinor Bose-Einstein condensate spontaneously acquires an antiferromagnetic pseudospin texture, which breaks the lattice symmetry similar to a supersolid.

Figure 1

Realization of orbital pseudospins in a superlattice. (a) The unit cell of the superlattice is a double well with offset $\mathrm{\Delta }$ and tunneling $J$. The two lowest eigenstates (pseudospin up and down) are coupled via a two-photon Raman process. (b) Raman process in the band structure of the superlattice. The ground state with quasimomentum $q=0$ is coupled to the edge of the Brillouin zone $q=(\pi /d)$ of the first excited band. (c) Top view of the superlattice with period $d={\lambda }_{\mathrm{IR}}/2=532\mathrm{nm}$. Raman coupling is implemented by two ${\lambda }_{\mathrm{IR}}$ beams: one along the superlattice ($z$ direction), the other along the $x$ direction. SOC (curved arrows) transfers transverse recoil in the $x$ direction to the atoms (dashed arrows).

Figure 2

Spontaneous formation of an antiferromagnetic spin texture. (a),(b) Time-of-flight (TOF) pattern of atoms in the ground (first excited) band of the superlattice. After preparation of the $|\uparrow ⟩$ state with quasimomentum $q=0$, it relaxes to the bottom of the band at $q=\pi /d$. (c) An equal mixture of spin states is prepared by rapidly switching the superlattice parameters. The two spin states can be separated in the TOF by a pseudospin Stern-Gerlach effect [23]. The figure shows that both spin states are in $q=0$ before the relaxation. (d) After relaxation, spinor BECs with states $|\downarrow ⟩$, $q=0$ and $|\uparrow ⟩$, $q=\pi /d$ are observed. The momentum pattern implies a periodic structure at $2d$, twice the lattice constant, indicating that an antiferromagnetic spin structure with a doubled unit cell has formed. The plus and minus signs indicate (one possible choice for) the phase of the BEC wave function. $n$ is the site index.

Figure 3

Characterization of spinor BECs through their momentum distributions. (a),(e) TOF images of the $|\downarrow ⟩$ and $|\uparrow ⟩$ states, respectively. (b),(f) Schematics of the momentum peaks for $|\downarrow ⟩$ and $|\uparrow ⟩$ with Raman coupling. Both the SOC (solid arrows) and the density modulation (dashed arrows) are shown. The main peak (filled circle) is equal to the quasimomentum of the state. Extra peaks (open circles) appear due to the periodic potential. (c),(d),(g),(h) Same as in (a) and (e), but now with Raman coupling at different detunings $\delta$. The momentum components created by the Raman process are vertically shifted compared to (a) and (e) due to the transverse momentum kick. The momentum shift along the superlattice ($z$ direction) reflects the $\pi /d$ quasimomentum of the Raman lattice. The off-resonant density modulation creates momentum peaks that are symmetric along $+x$ and $-x$ [(c) and (g)], whereas resonant spin-orbit coupling creates unidirectional momentum transfer resulting in asymmetry[(d) and (h)]. (i) Spin-orbit coupled BEC with equal population in the spin up and spin down states.

Figure 4

Spin-orbit coupling resonances. Shown is the population imbalance between the “$+x$” and “$-x$” momentum peaks versus Raman detuning for the $|\downarrow ⟩\to |\uparrow ⟩$ (blue) and $|\uparrow ⟩\to |\downarrow ⟩$ (red) processes. The two sets of data were measured for the same superlattice parameters $V_{\mathrm{IR}}=7.5(2)E_r$, $V_{\mathrm{Gr}}=20(2)E_r$, and ${\varphi }_{\mathrm{SL}}\approx 0.22(1)\pi$, which gives $\mathrm{\Delta }\approx 37(1)\mathrm{kHz}$. The spin-orbit coupling strength $\beta$ was calculated to be 0.40(5) kHz. The solid lines are Gaussian fits to the resonances centered at 32.2(3) and 43.2(3) kHz. The Gaussian profile of the IR lattice inhomogeneously broadens the resonances. The error bars represent $1\sigma$ statistical uncertainty. Inset: resonance center frequencies versus the IR lattice depth $V_{\mathrm{IR}}$ for fixed ${\varphi }_{\mathrm{SL}}$. The resonances are linear in $V_{\mathrm{IR}}$ with a constant split equal to twice the recoil energy. The slope of the linear fit reveals ${\varphi }_{\mathrm{SL}}$. The error bars are the uncertainties of the fit.