Multifrequency optical lattice for dynamic lattice-geometry control

Ultracold atoms in optical lattices are pristine model systems with a tunability and flexibility that goes beyond solid-state analogies, e.g., dynamical lattice-geometry changes allow tuning a graphene lattice into a boron-nitride lattice. However, a fast change of the lattice geometry remains intrinsically difficult. Here we introduce a multifrequency lattice for fast and flexible lattice-geometry control and demonstrate it for a three-beam lattice, realizing the full dynamical tunability between honeycomb lattice, boron-nitride lattice, and triangular lattice on the microsecond scale, i.e., fast compared to the relevant energy scales. At the same time, the scheme ensures intrinsically high stability of the lattice geometry. We introduce the concept of a geometry phase as the parameter that fully controls the geometry and observe its signature as a staggered flux in a momentum space lattice. Tuning the geometry phase allows us to dynamically control the sublattice offset in the boron-nitride lattice. We use a fast sweep of the offset to transfer atoms into higher Bloch bands and perform a new type of Bloch band spectroscopy by modulating the sublattice offset. Finally, we generalize the geometry phase concept and the multifrequency lattice to three-dimensional optical lattices and quasiperiodic potentials. This scheme will allow further applications such as novel Floquet and quench protocols to create and probe, e.g., topological properties.

Figure 1

Realization of a hexagonal lattice with tunable and stable geometry as multifrequency lattice. (a) In the multifrequency design a hexagonal lattice is realized by superimposing three 1D lattices of different laser frequencies indicated as yellow (${\nu }_a$), green (${\nu }_b$), and blue (${\nu }_c$). Bold arrows represent carriers, thin arrows represent sidebands of the three lattice beams with wave vectors ${\mathit{k}}_1$, ${\mathit{k}}_2$, and ${\mathit{k}}_3$. (b) The beam configuration in panel (a) can be obtained by appropriate sidebands (${\nu }_{\alpha }$, ${\nu }_{\beta }$, ${\nu }_{\gamma }$) imprinted on the laser beams together with suitable detunings between the beams. (c) The resulting lattice can be freely tuned between a honeycomb lattice (center), boron-nitride lattices, and triangular lattice (edges). The numbers below the plots indicate the corresponding geometry phase, the hexagons show a unit cell. (d) Clouds of 30 000 ${}^{87}\mathrm{Rb}$ atoms in different lattice potentials $[{\varphi }_{\mathrm{g}}/\left(2\pi \right)=0,0.005,0.03$, from left to right].

Figure 2

Quantum walk in a triangular momentum space lattice with staggered flux showing inversion-symmetry breaking. (a) A 1D optical lattice couples momentum modes separated by its wave vector $\mathit{b}$ (red arrows). The coupling becomes increasingly off-resonant for higher momentum modes (black arrows). (b) The resulting lattice in momentum space has a staggered flux $\mathrm{\Phi }$ resulting from the Peierls phases for the directions indicated by the arrows. The Peierls phases and the corresponding arrows are displayed in the same color. (c) Imbalance $\mathcal{I}=\frac{p_a-p_o}{p_a+p_o}$ between the first-order peak populations along ($p_a$) and opposite ($p_o$) to the reciprocal lattice vectors ${\mathit{b}}_i$ as a function of the geometry phase. The error bars denote the standard deviation of the mean of the three equivalent pairs. The insets show the raw images for the points indicated by the arrows. The images were taken for an effective pulse time of 4.7 $\mu \mathrm{s}$. (d) Top panel, time evolution of the root mean square width of the distribution of experimental data (symbols) and numerics (lines) for ${\varphi }_{\mathrm{g}}/2\pi =0$ (dark blue, circles), ${\varphi }_{\mathrm{g}}/2\pi =0.25$ (middle blue, triangles), and ${\varphi }_{\mathrm{g}}/2\pi =0.5$ (light blue, crosses). The external trap leads to halting of the expansion, with different exact behavior as a function of the staggered flux. Exemplary distributions at a pulse time of 16.4 $\mu \mathrm{s}$ (indicated by the arrow) show the rich dynamics in dependence of the staggered flux (lower panels). The lines in panels (c) and (d) are numerically calculated using $V=6.5E_{\mathrm{rec}}$ (Appendix pp8).

Figure 3

Preparation of higher bands via sweeps of the geometry phase. (a) 1D cuts through the potentials before and after the sweeps inverting the energy offset between the A and B sites ${\mathrm{\Delta }}_{\mathrm{AB}}$. The populated orbitals are highlighted in red. The four final situations correspond to the four panels in (b). (b) Band mapping images for linear sweeps from ${\varphi }_{\mathrm{g}}/\left(2\pi \right)=-0.048$ to final values of ${\varphi }_{\mathrm{g}}/\left(2\pi \right)=0.048$, 0.064, 0.095, and 0.153 within $60\mu \mathrm{s}$ ($20\mu \mathrm{s}$ in the last image) followed by $500\mu \mathrm{s}$ hold time. The atoms are transferred to the 2nd band, 2nd+3rd+4th bands (at the band touching), 4th band, and 7th band, respectively. Below the Brioullin zones corresponding to these bands are shown (c) Band structure of the hexagonal lattice as a function of ${\varphi }_{\mathrm{g}}$. The band structure is symmetric under reflection around ${\varphi }_{\mathrm{g}}=0$ and ${\varphi }_{\mathrm{g}}=\pi$ (not shown). The lowest band gap given by ${\mathrm{\Delta }}_{\mathrm{AB}}$ increases linearly with ${\varphi }_{\mathrm{g}}$ up to about ${{\varphi }_{\mathrm{g}}}^{\mathrm{sp}}$, where the energetically highest $s$ orbital is degenerate with the lowest $p$ orbital (${{\varphi }_{\mathrm{g}}}^{\mathrm{sp}}/\left(2\pi \right)=0.064$ for our lattice depth of $V=6.3E_{\mathrm{rec}}$). Via a series of (avoided) crossings, the bands rearrange from a honeycomb lattice with two $s$ bands (${\varphi }_{\mathrm{g}}=0$) to a triangular lattice (${\varphi }_{\mathrm{g}}=\pi$) with a single $s$ band. Arrows indicate the final values of ${\varphi }_{\mathrm{g}}$ of the sweeps stated in panels (b). (d) Plots of the band structures at ${\varphi }_{\mathrm{g}}/\left(2\pi \right)=0$, 0.032, 0.064, and 0.134, the latter two featuring triple band crossings.

Figure 4

Excitation spectra of a boron-nitride lattice with ${\varphi }_{\mathrm{g}}/\left(2\pi \right)=0.024$ for different modulation types. Relative population of the lowest band obtained via band mapping versus spectroscopy frequency ${\nu }_{\mathrm{sp}}$ for sublattice modulation (dark blue, circles), amplitude modulation (middle blue, triangles) and circular lattice shaking (light blue, crosses). The lines are spline interpolations as a guide to the eye. The modulation indices are 0.02, 0.05, and 600 $\mathrm{Hz}/{\nu }_{\mathrm{sp}}$, respectively, chosen to yield good signal at the modulation time of 10 ms. The red shaded areas indicate the expected transition frequencies into the stated bands for a lattice depth of $V_{\mathrm{latt}}=9.3E_{\mathrm{rec}}$. The width of the areas indicates the spectrum of quasimomentum-preserving transition frequencies starting from a full lowest band.

Figure 5

Proposal for a tunable 3D lattice of trimers via multifrequency design. (a) $V_{13}\left(\mathit{r}\right)$ and $V_{46}\left(\mathit{r}\right)$ for $z=0,\lambda /6,\lambda /3$, as defined in the main text. The geometry phase (${\varphi }_{g,1}/\left(2\pi \right)=0.29$, ${\varphi }_{g,2}={\varphi }_{g,3}={\varphi }_{g,1}/3$) is chosen such that the minima of $V_{46}\left(\mathit{r}\right)$, found at $z=m\lambda /3$ (with $m$ integer), coincide with a maximum of $V_{13}\left(\mathit{r}\right)$ (solid and dashed circle). The relative depth of the two potentials is given by $V_4=1.8V_1$. (b) Potential landscape plotted for different sections parallel to the $xy$ plane of size $2.2\lambda \times 2.2\lambda$. The color map is cropped to emphasize the relevant potential minima. The lattice has three sublattices, i.e., trimers in the $xy$ plane, and tunneling along $z$ couples lattice sites in different sublattices. (c) Potential in the plane of the trimers with different potential offsets between the sublattices, engineered by shifting $V_{46}\left(\mathit{r}\right)$ with respect to $V_{13}\left(\mathit{r}\right)$ in the $xy$ plane using different choices of the geometry phases ${\varphi }_{g,2}$ and ${\varphi }_{g,3}$ [compare Fig. 5]. A balanced situation is achieved without shift (first panel) and sublattice offsets are reached by shifting along the $x$ direction (second panel) and the $y$ direction (third panel).

Figure 6

Geometry phase in a quasicrystal lattice. (a) Lattice potential of the form (Eq. (B1)) as a function of ${\varphi }_{\mathrm{g}}$ in an area of $10\lambda \times 10\lambda$. The lattice potential is normalized to the interval [0,1]. (b) Orientation of the reciprocal lattice wave vectors ${\mathit{b}}_i$, with $|{\mathit{b}}_i|\sim 1.9\frac{2\pi }{\lambda }$. In momentum space, pentagonal plaquettes with flux $\mathrm{\Phi }=\pm ({\varphi }_{\mathrm{g}}+\pi )$ can be found. (c) Histograms of the values of the normalized potential landscape for ${\varphi }_{\mathrm{g}}/\left(2\pi \right)=0,1/4,1/2$. Orange lines are drawn to mark the slope at the smallest potential values, i.e., in the relevant region for the lowest lying states. The slope is largest at ${\varphi }_{\mathrm{g}}=0$ because of the larger number of lattice sites of similar energy. (d) Zoom in ($2.58\lambda \times 2.58\lambda$) on one of the 10-fold symmetric pattern that can be found for ${\varphi }_{\mathrm{g}}=0$, demonstrating the breaking of the symmetry between the odd and the even minima of the structure for finite values of ${\varphi }_{\mathrm{g}}$. (e) Normalized quasicrystal potential in a $5\lambda \times 5\lambda$ area as a function of a phasonic degree of freedom $p$: $V_{\mathrm{pot}}\left(\mathit{r}\right)={\sum }_i^{5}cos({\mathit{b}}_i·\mathit{r}+pcos\left(\frac{4\pi }{5}i\right)+\frac{\pi }{3})$, with ${\varphi }_{\mathrm{g}}=5·\frac{\pi }{3}=\frac{-\pi }{3}$. The potential pattern remains self-similar. This degree of freedom does not couple to ${\varphi }_{\mathrm{g}}$ and to translations in the physical space. (f) One of the 14-fold symmetric pattern that can be found for ${\varphi }_{\mathrm{g}}=0$ in the 7-fold symmetric configuration with $N=7$ (system size $3.8\lambda \times 3.8\lambda$). Here $|{\mathit{b}}_i|=1.95\frac{2\pi }{\lambda }$, as obtainable with seven beams by interference of every beam pair with $\pm \frac{6\pi }{7}$ relative angle.

Figure 7

Implementation of the multifrequency lattice with stable phases between the frequency components. The laser beam from a single source is split into three beams using $\lambda /2$ waveplates (not shown) and polarising beam splitters. Each beam has an EOM to add a sideband and an AOM to shift the frequency appropriately before coupling into separate optical fibers. The RFs for the EOMs are derived from a single source and one frequency is derived as the difference frequency of the other two using a mixer and a low pass filter. This setup produces the spectra sketched in Fig. 1.

Figure 8

Different methods for the calibration of the geometry phase. (a) Relative strengths of imbalanced Bragg-peaks across all geometries. Around ${\varphi }_{\mathrm{g}}=0$ (honeycomb) the Bragg-peak populations are much more sensitive to the lattice imbalance (see text). The center is determined by the average of three heuristic Gaussian fits to the data, resulting in a 68% confidence interval of the fit of $\delta {\varphi }_{\mathrm{rf}}^0=0.00153\times 2\pi =9.58\mathrm{mrad}$. The symbols correspond to experimental data, the lines to the fits. The color encodes the direction in momentum space. (b) Spectroscopy measurement closely around ${\varphi }_{\mathrm{g}}=0$ followed by band mapping, which yields the relative population of the 2nd band (symbols). A spectroscopy frequency resonant to a small AB offset is used, resulting in two distinctive peaks at opposite sites of ${\varphi }_{\mathrm{g}}=0$. The center is determined by a heuristic three Gaussian fit with forced equal distances to each other (line). The 68% confidence interval of the fit is then $\delta {\varphi }_{\mathrm{rf}}^0=0.00016\times 2\pi =0.98\mathrm{mrad}$. (c) Real-space measurement closely around ${\varphi }_{\mathrm{g}}=0$, with relative A-site (blue symbols) and B-site populations (red symbols). The error bars denote the standard deviation from two to three iterations. The populations do not add up to one due to finite occupations attributed to the spaces between the lattice sites. Via a heuristic fit of two linear curves (lines, valid only up to small offsets) the point of equal population ${\varphi }_{\mathrm{g}}=0$ is determined with a 68% confidence interval of the fit of $\delta {\varphi }_{\mathrm{rf}}^0=0.00012\times 2\pi =0.75\mathrm{mrad}$.

Figure 9

Evolution of the geometry phase calibration over four months. Every point results from a real-space phase calibration as in Fig. 8 with the error bar given by the 68% confidence interval of the corresponding fit.

Figure 10

Quantum walk in a triangular momentum space lattice. (a) ${\varphi }_{\mathrm{g}}=0$, (b) ${\varphi }_{\mathrm{g}}=\pi /2$, (c) ${\varphi }_{\mathrm{g}}=\pi$. Each subfigure contains 10 different hold times, starting from $3.9\mu \mathrm{s}$ and increasing in steps of $5\mu \mathrm{s}$ up to $48.9\mu \mathrm{s}$. The lattice depth is the same as in Fig. 2.

Figure 11

Band-resolved excitation spectra of a boron-nitride lattice with ${\varphi }_{\mathrm{g}}/\left(2\pi \right)=0.024$. Relative populations of the different bands (see labels of the panels) versus spectroscopy frequency for sublattice modulation (dark blue, circles), amplitude modulation (middle blue, triangles) and circular lattice shaking (light blue, crosses). The shaded areas indicate the expected transition frequencies into the stated bands for single-photon transitions (red) and two-photon transitions (light red). The width of the areas indicates the spectrum of quasimomentum-preserving transition frequencies starting from a full lowest band. The parameters for the data set are stated in Fig. 4.

Figure 12

Resonant excitation strength from the lowest band. Transition matrix elements into higher bands for sublattice modulation (dark blue, circles), amplitude modulation (middle blue, triangles), and circular lattice shaking (middle blue, triangles) integrated over the first Brillouin zone plotted on a logarithmic scale. The examined geometries are from left to right: ${\varphi }_{\mathrm{g}}/\left(2\pi \right)=0,0.024,0.5$. The modulation strengths correspond to the experimental values of Fig. 4, i.e., ${\epsilon }_{\mathrm{sm}}=0.02$ for sublattice modulation, ${\epsilon }_{\mathrm{am}}=0.05$ for amplitude modulation and ${\epsilon }_{\mathrm{cs}}=0.08$ (resonance to 2nd band), 0.035 (resonance to 3rd and 4th bands), 0.028 (resonance to 5th and 6th bands), 0.021 (theoretical expectation for resonance to 7th band) for circular lattice shaking.

Figure 13

PCA analysis of distorted higher BZs. (a) Example band mapping images of atoms in higher bands indicate distortions relative to the ideal BZs (black lines). (b) Masks of the first five distorted BZs obtained by the PCA analysis of the band mapping images. (c) The first nine ideal BZs sampled with the same pixel size for comparison. (d) The first nine PCs of the data set of 147 images as in (a) carry information on the distorted BZs.