Probing the flat band of optically trapped spin-orbital-coupled Bose gases using Bragg spectroscopy

Motivated by recent efforts in creating flat bands in ultracold atomic systems, we investigate how to probe a flat band in an optically trapped spin-orbital-coupled Bose-Einstein condensate using Bragg spectroscopy. We find that the excitation spectrum and the dynamic structure factor of the condensate are dramatically altered when the band structure exhibits various levels of flatness. In particular, when the band exhibits perfect flatness around the band minima corresponding to a near-infinite effective mass, a quadratic dispersion emerges in the low-energy excitation spectrum; in sharp contrast, for the opposite case when an ordinary band is present, the familiar linear dispersion arises. Such linear-to-quadratic crossover in the low-energy spectrum presents a striking manifestation of the transition of an ordinary band into a flat band, thereby allowing a direct probe of the flat band by using Bragg spectroscopy.

Figure 1

On the left Bragg spectroscopy probes a single-particle flat band generated by an optically trapped spin-orbit-coupled Bose gas. Shown on the right is the schematic setup for implementing a flat band realized in 1D spin-orbit-coupled Bose gas proposed by Ref. [9]. Here ${\Omega }_1$ and ${\Omega }_2$ are the Rabi frequencies of the Raman lasers for generating SOC. The interference of other two counterpropagating laser beams labeled by ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$ generates an optical lattice. Bragg spectroscopy can be described theoretically by the dynamic structure factor of the model system.

Figure 2

(a1)–(d1) Lowest Bloch bands of an optically trapped spin-orbit-coupled BEC (in the inset, the lowest two Bloch bands are plotted). (a2)–(d2) Lowest three Bogoliubov bands ${\omega }_j$ and the vertical lines represent the excitation strengths $Z_j$ ($j=1,2,3$) toward the first three bands for $q=0.8k_L$, $-1.2k_L$, and $2.8k_L$, respectively. (a3)–(d3) Lowest excitation spectrum. (a4)–(d4) Excitation strength $Z_{1,2,3}$ and static structure factor $S$ via a given transferred momentum $q$. Here $m^{*}$ labels the effective mass at the energy minima $k_{\min }$. The parameters are given as follows: $c=0.0500$ and (a1)–(a4) $\gamma =0.5000$, $V_0=0.6000$, and $\Omega =0.8000$; (b1)–(b4) $\gamma =1.0500$, $V_0=1.0000$, and $\Omega =1.1500$; (c1)–(c4) $\gamma =0.6100$, $V_0=2.0000$, and $\Omega =0.3700$; and (d1)–(d4) $\gamma =0.7000$, $V_0=1.0000$, and $\Omega =0.5005$.