Topologically Induced Avoided Band Crossing in an Optical Checkerboard Lattice

We report on the condensation of bosons in the 4th band of an optical checkerboard lattice providing a topologically induced avoided band crossing involving the 2nd, 3rd, and 4th Bloch bands. When the condensate is slowly tuned through the avoided crossing, accelerated band relaxation arises and the zero momentum approximately $C4$-invariant condensate wave function acquires finite momentum order and reduced $C2$ symmetry. For faster tuning Landau-Zener oscillations between different superfluid orders arise, which are used to characterize the avoided crossing.

Figure 1

(a) The lattice comprises two classes of sites with different well depth denoted by $\mathcal{A}$ and $\mathcal{B}$. The gray area denotes the Wigner-Seitz unit cell. (b) The first BZ with a path marked connecting the $\Gamma$, $X$, and $M$ points, respectively. (c) Energies of the 2nd ($B_2$), 3rd ($B_3$), and 4th ($B_4$) Bloch bands plotted versus $\Delta V$ for a lattice with perfect $C4$ symmetry for $V_0=7.0E_{\mathrm{rec}}$ and $\Delta V_c=6.1E_{\mathrm{rec}}$. The detail on the lower edge shows a plot of these bands for $\Delta V=\Delta V_c$ within the 1st BZ along the trajectory illustrated in (b). (d) The bands of (c) are replotted for a lattice with weakly broken $C4$ symmetry for $V_0=7.8E_{\mathrm{rec}}$ and $\Delta V_c=6.1E_{\mathrm{rec}}$. The details on the lower edge show plots analogous to that shown in  (c), however, for $\Delta V=\Delta V_c\pm E_{\mathrm{rec}}$. The (thin, gray) dash-dotted lines mark the energy of the 4th band at the $\Gamma$ point.

Figure 2

Momentum spectra of the $B_4$ band condensate are shown in (a) and (b) for $(\Delta V-\Delta V_c)/E_{\mathrm{rec}}=2.9$ and in (c) and (d) for $(\Delta V-\Delta V_c)/E_{\mathrm{rec}}=9.9$, respectively. The experimental results in (a) and (c) are compared to calculations in (b) and (d). The $\Delta V$ dependence of the populations of the Bragg peaks identified in the dashed black rectangle in (d) is shown in (e) (experimental observations) and (f) (calculations).

Figure 3

Band mapping plots recorded for $(\Delta V-\Delta V_c)/E_{\mathrm{rec}}=4.5$ and holding times $20\mu \mathrm{s}$, 20 ms, 100 ms. In the lower right corner a map of the theoretical BZs $1,2,3,4$ is plotted. The black circles indicate the observed condensation points at the energy minima of the $B_4$ band, which connect to the center of the 1st zone via reciprocal lattice vectors (white arrow). (b) The decay time of the 4th (blue disks) and the refilling time of the 1st band (black diamonds) are plotted versus $\Delta V$. The black line plots show the scaled overlap integrals $I_1$ (solid), $I_2$ (dashed), $I_3$ (dash-dotted) defined in Eq. (2).

Figure 4

(a) Momentum spectra (left, observation; right, calculation) on the left side of the anticrossing ($\Delta V=\Delta V_c-1.3E_{\mathrm{rec}}$). A sufficiently large gap ($\approx 0.8E_{\mathrm{rec}}$) was adjusted, in order to prevent Landau-Zener dynamics. (b) Landau-Zener dynamics for a gap of $\approx 0.26E_{\mathrm{rec}}$. The visibility of the $\pm (1,-1)\hslash k$ Bragg peaks (blue disks and green triangles) and that of the $(0,0)\hslash k$ peak (red squares) are plotted versus the holding time after rapid tuning over the anticrossing to $(\Delta V-\Delta V_c)/E_{\mathrm{rec}}=-3.0$. (c) Fourier spectrum of the oscillating $(-1,1)\hslash k$ peak with first and second harmonic components indicated by black arrows. (d) Blue disks denote the first harmonic frequencies derived from plots as in (b). The solid line shows the calculated energy difference between the 4th and the 2nd band.