Experimental realization of spin-tensor momentum coupling in ultracold Fermi gases

We experimentally realize spin-tensor momentum coupling (STMC) using three ground Zeeman states coupled by three Raman laser beams in an ultracold atomic system of ${}^{40}\mathrm{K}$ Fermi atoms. This type of STMC consists of two bright-state bands as a spin-orbit coupled spin-1/2 system and one dark-state middle band. Using radio-frequency spin-injection spectroscopy, we investigate the energy band of STMC. It is demonstrated that the middle state is a dark state in the STMC system. The experimental realization of STMC open the door for further exploring exotic quantum matter.

Figure 1

Schematics of the Raman lasers configuration and atomic levels of generating STMC. (a) Raman lasers configuration to generate STMC in ultracold Fermi gases. (b) Raman transitions among three hyperfine spin states with detuning $\delta$. (c1) and (c2) Theoretical single-particle band structure for Raman strength $\hslash {\mathrm{\Omega }}_R$=$1.0E_r$ and $2.5E_r$, respectively. The detuning $\hslash \delta$ is set as $0.1E_r$. The lowest band indicates eigenstate $|\alpha \rangle$, the highest band indicates eigenstate $|\beta \rangle$, and the middle one indicates eigenstate $|\gamma \rangle$.

Figure 2

Energy-band structure of 1D SOC ultracold Fermi gases. (a) A pair of Raman beams couple two spin-states $|9/2,1/2\rangle (|\uparrow \rangle )$ and $|9/2,-1/2\rangle (|0\rangle )$ to generate the 1D SOC system with the Raman coupling strength $\hslash {\mathrm{\Omega }}_R=2.5E_r$ and the detuning $\hslash \delta =0E_r$. (b) Time-of-flight (TOF) absorption image of spin-injection spectroscopy at a given frequency of rf field. (c1) and (c2) Reconstructed momentum- and spin-resolved $|\uparrow \rangle$ (blue) and $|0\rangle$ (red) spectra, respectively, when driving atoms from the free spin-state $|9/2,3/2\rangle$. (c3) Displaying two graphs (c1) and (c2) simultaneously.

Figure 3

Energy-band structure of STMC ultracold Fermi gases. (a) Schematic of the process of spin-injection spectroscopy with preparing in the free spin-state $|9/2,3/2\rangle$. The vertical arrows represent the transitions driven by the rf field. The frequency of the rf field is scanned. (b) TOF image of spin-injection spectroscopy at a given frequency of the rf field. (c1)–(c3) Reconstructed spin-resolved $|\uparrow \rangle$ (red), $|0\rangle$ (green), and $|\downarrow \rangle$ (blue) spectra, respectively, when driving atoms from the free spin-state $|9/2,3/2\rangle$ to the STMC system with the Raman coupling strength $\hslash \mathrm{\Omega }=2.5E_r$ and the detuning $\hslash \delta =0E_r$. (c4) Displaying the three graphs (c1)–(c3) simultaneously. (d1)–(d4) spin-injection spectroscopy with the values of $\hslash \mathrm{\Omega }=3E_r$ and $\hslash \delta =0E_r$.

Figure 4

Another method to measuring momentum-resolved rf spectroscopy of the STMC ultracold Fermi gases. (a) Time-of-flight absorption image of rf spectroscopy at a given frequency of the rf field. (b) Schematic of the process of rf spectroscopy with initially preparing atoms in STMC. The vertical arrows represent the transitions driven by the rf field. (c) Reconstructed single-particle dispersion and atom population when transferring atoms from the STMC ultracold Fermi gases system to free spin-state $|9/2,3/2\rangle$ for $\hslash \mathrm{\Omega }=2.5E_R$ and the detuning $\hslash \delta =0E_R$.