Observation of Floquet bands in driven spin-orbit-coupled Fermi gases

Periodic driving of a quantum system can significantly alter its energy bands and even change the band topology, opening a completely new avenue for engineering novel quantum matter. Although important progress has been made recently in measuring topological properties of Floquet bands in different systems, direct experimental measurement of full Floquet band dispersions and their topology change is still demanding. Here we directly measure Floquet band dispersions in a periodically driven spin-orbit-coupled ultracold Fermi gas using spin-injection radio-frequency spectroscopy. We observe that the Dirac point originating from two-dimensional spin-orbit coupling can be manipulated to emerge at the lowest or highest two dressed bands by fast modulating Raman laser frequencies, demonstrating topological change of Floquet bands. Our work will provide a powerful tool for understanding fundamental Floquet physics as well as engineering exotic topological quantum matter.

Figure 1

(a) Energy-level diagram of Fermi gases ${}^{40}\mathrm{K}$. Three hyperfine spin states are coupled with the electronic excited states through three Raman lasers. The atoms are initially prepared in the free reservoir spin state $|9/2,5/2\rangle$. (b) Configuration of three Raman lasers in the $xy$ plane. The detunings of the Raman lasers 2 and 3 are modulated as ${\delta }_{2\left(0\right)}+{\delta }_{m}cos\left(\omega t\right)$ and ${\delta }_{3\left(0\right)}+{\delta }_{m}cos(\omega t+{\varphi }_0)$. (c) Plot of the product of the effective Raman coupling strengths $\eta ={\mathrm{\Omega }}_{12}^{\prime }{\mathrm{\Omega }}_{13}^{\prime }{\mathrm{\Omega }}_{23}^{\prime }$ (scaled by ${\eta }_0={\mathrm{\Omega }}_{12}{\mathrm{\Omega }}_{13}{\mathrm{\Omega }}_{23}$) as a function of the modulation parameter ${\delta }_m/\omega$. The background colors indicate the sign of $\eta /{\eta }_0$ which determines the position of the Dirac point. (d) Band structures near the Dirac points for different modulation parameter ${\delta }_m/\omega$. Only two bands that exhibit a Dirac point are shown, with the lower, middle, and higher bands labeled by L, M, and H, respectively. The relative phase ${\varphi }_0=\pi /2$ in (c) and (d).

Figure 2

Observation of Floquet band topological change. The upper and middle panels represent theoretically calculated and experimentally measured 2D Floquet band dispersions, respectively. The red dots represent the Dirac points. The lower panel represents corresponding 1D dispersions along (a)–(d) $k_y=0.05k_r$ and (e) $k_y=-0.7k_r$, where the grayness of the lines indicates the population of the final state $\left|3\right.〉$. The modulation amplitudes of the Raman detunings are (a) ${\delta }_m/\omega =0$, (b) 1.03, (c) 1.695, (d) 2.416, and (e) 3.3. Other parameters are ${\mathrm{\Omega }}_{12}=5.46E_r$, ${\mathrm{\Omega }}_{13}=4.62E_r$, ${\mathrm{\Omega }}_{23}=-4.2E_r$, ${\delta }_{2\left(0\right)}=-2.47E_r$, and ${\delta }_{3\left(0\right)}=0.93E_r$. The relative phase is ${\varphi }_0=\pi /2$. Note that the orientation of the $k_x$ axes is different for a better view of the band structures.

Figure 3

(a) Energy separations between three energy bands at ${\mathbf{k}}_0$, i.e., the position of the Dirac point in the absence of modulation. The symbols and solid lines correspond to experimental and theoretical results, respectively. (b) Change of the position of the Dirac point as a function of the modulation parameter ${\delta }_m/\omega$. The magenta and cyan lines correspond to a theoretical plot of $k_x$ and $k_y$ of the Dirac point. The symbols correspond to the positions of the experimentally measured Dirac points. All other parameters are the same as Fig. 2. The background colors in both panels indicate whether the Dirac point exhibits at the crossing of the two lower (green) or the two higher (yellow) bands.

Figure 4

Effect of the initial relative phase ${\varphi }_0$. The format and parameters are the same as Fig. 3, except that (a),(b) ${\varphi }_0=0$ and (c),(d) ${\varphi }_0=\pi$, respectively. For ${\varphi }_0=0$, the Raman coupling ${\mathrm{\Omega }}_{23}$ is not modulated and the Dirac point remains at the crossing of two lower bands.

Figure 5

Schematic of generating the Raman lasers. $\lambda /2$: half-wave plate; $\lambda /4$: quarter-wave plate; PBS: polarized beam splitter; AOM: acousto-optic modulator.