Spin-Injection Spectroscopy of a Spin-Orbit Coupled Fermi Gas

The coupling of the spin of electrons to their motional state lies at the heart of recently discovered topological phases of matter. Here we create and detect spin-orbit coupling in an atomic Fermi gas, a highly controllable form of quantum degenerate matter. We directly reveal the spin-orbit gap via spin-injection spectroscopy, which characterizes the energy-momentum dispersion and spin composition of the quantum states. For energies within the spin-orbit gap, the system acts as a spin diode. We also create a spin-orbit coupled lattice and probe its spinful band structure, which features additional spin gaps and a fully gapped spectrum. In the presence of $s$-wave interactions, such systems should display induced $p$-wave pairing, topological superfluidity, and Majorana edge states.

Figure 1

Realization of spin-orbit coupling in an atomic Fermi gas. (a) Energy bands as a function of quasimomentum $q$ for Raman coupling strength of $\hslash {\Omega }_R=0.25E_R$ and $\hslash \delta =0E_R$. Energy bands for $\hslash {\Omega }_R=\hslash \delta =0E_R$ are shown with dashed lines. Color indicates spin composition of the states. (b) A pair of Raman beams at $\pm 19°$ relative to the $\widehat{y}$ axis couples states $|\downarrow ,k_x=q⟩$ and $|\uparrow ,k_x=q+Q⟩$. A bias magnetic field $\mathbf{B}$ in the $\widehat{z}$ direction provides the quantization axis. (c) Energy level diagram of states coupled by the Raman fields: $\hslash \delta$ is the two-photon detuning. The hyperfine interaction splits $|\uparrow ⟩$ and $|\downarrow ⟩$ by $\hslash {\omega }_0$, and the relevant polarization components are $\pi$ and ${\sigma }_{+}$. (d) Momentum-dependent Rabi oscillations for $\hslash {\Omega }_R=0.71(2)E_R$ and $\hslash \delta =-0.25(1)E_R$. Atoms are prepared in $|\downarrow ⟩$ (red) and are subsequently projected into a superposition of eigenstates as the Raman field is pulsed on. (e) A $\pi$ pulse for the resonant momentum class is applied at different $\hslash \delta$ for $\hslash {\Omega }_R=0.035(5)E_R$. (f),(g) Adiabatic loading and unloading of atoms into the upper (lower) band with $\hslash {\Omega }_R=0.53(5)E_R$. The Raman beams are turned on with $\delta =\mp 8.5{\Omega }_R$, which is then swept linearly to $\delta =0$ and back at a rate of $|\dot{\delta }|=0.27(5){\Omega }_R^2$.

Figure 2

Spin-injection spectroscopy. (a) An rf pulse injects atoms from the reservoir states (shown in black) $|\uparrow {⟩}_R$ and $|\downarrow {⟩}_R$ into the spin-orbit coupled system (shown in red and blue). Injection occurs when the rf photon energy equals the energy difference between the reservoir state and the spin-orbit coupled state at quasimomentum $q$. (b),(c) Spin-resolved $|\downarrow ⟩$ and $|\uparrow ⟩$ spectra, respectively, when transferring out of $|\uparrow {⟩}_R$. Here, $\hslash {\Omega }_R=0.43(5)E_R$ and $\hslash \delta =0.00(3)E_R$. (d),(e) Spin-resolved $|\downarrow ⟩$ and $|\uparrow ⟩$ spectra, respectively, when transferring out of $|\downarrow {⟩}_R$ for the same Raman strength $\hslash {\Omega }_R$. (f), (g), and (h) The reconstructed spinful dispersions for $\hslash \delta =0.00(3)E_R$ and $\hslash {\Omega }_R=0E_R$, $\hslash {\Omega }_R=0.43(5)E_R$, and $\hslash {\Omega }_R=0.9(1)E_R$, respectively.

Figure 3

Creating and probing a spin-orbit coupled lattice. (a) The addition of a radio frequency field allows momentum transfer of any multiple of $Q$, producing a spinful lattice band structure. (b) The band structure of the Raman-rf system in the repeated zone scheme. Band structures from top to bottom correspond to $\hslash {\Omega }_{\mathrm{RF}}=0$ and $\hslash {\Omega }_R=0.25E_R$, $\hslash {\Omega }_R=0.5E_R$ and $\hslash {\Omega }_{\mathrm{RF}}=0$, and $\hslash {\Omega }_R=0.5E_R$ and $\hslash {\Omega }_{\mathrm{RF}}=0.25E_R$. In the bottommost band, all degeneracies are lifted. (c) Spin injection from free particle bands to spinful lattice bands, starting from $|\downarrow {⟩}_R$. Transitions near zero rf detuning ($h\Delta \nu \sim 0$) that give rise to dominant spectral features are identified. (d) Experimental spectrum of the Raman-rf system with $\hslash {\Omega }_R=0.40(5)E_R$ and $\hslash {\Omega }_{\mathrm{RF}}=0.28(2)E_R$ in the spin $|\downarrow ⟩$ channel after injection from reservoir $|\downarrow {⟩}_R$. (e) The theoretical spectra corresponding to (d). Features corresponding to the gaps and transitions identified in (c) are labeled.

Figure 4

Evolution of spin-textured energy bands of a spin-orbit coupled lattice. (a),(b) Experimental Raman-rf spin-injection spectra for injection from $|\downarrow {⟩}_R$ for channels $|\downarrow ⟩$ and $|\uparrow ⟩$, respectively. The color map used is the same as Figs. 2b, 2e after rescaling to the maximum intensity. Interaction effects between $|\uparrow ⟩$ with $|\downarrow {⟩}_R$ (see Fig. S4) makes only the dominant features resolvable in $|\uparrow ⟩$, while finer features are visible in $|\downarrow ⟩$. (c) Reconstructed band structure for $\hslash {\Omega }_R=0.93(7)E_R$ and $\hslash {\Omega }_{\mathrm{RF}}=0.28(2)E_R$. Color indicates the spin texture. (d)–(f) Experimentally measured spin components $S_y$ and $|S_z|$ as a function of momentum $k_x$ for the lattice wave functions corresponding to the bottommost band in (c).