Experimental realization of a high precision tunable hexagonal optical lattice

As an additional degree of freedom, different from charge and spin, orbital physics plays a central role for superconductivity, magnetism and transport in transition-metal-oxides due to two fundamental features, i.e., orbital degeneracy and orbital anisotropy[1]. In periodic crystals, orbitals describe the motion of electrons localized at a lattice site. Electron clouds with different symmetries can arise such as $s$, $p$, and $d$ orbitals, distinguished by their angular momentum. To simulate orbital degrees of freedom with ultracold atoms, one may apply artificial periodic potentials, called optical lattices, by interfering several laser beams, and prepare the atoms in higher Bloch bands. Wannier functions associated with the Bloch bands describe the character of the orbitals. As a versatile platform for quantum simulation [2, 3, 4, 5, 6, 7], ultracold atoms in optical lattices have become an ideal system for exploring orbital physics in electronic condensed matter. In contrast to the widely studied $s$ orbital physics in optical lattices [4, 8, 9], the interplay of orbital degeneracy and spatial anisotropy of higher orbitals can induce more complex many-body scenarios with remarkable new features [10].

Over the past decade, $p$ orbitals in optical lattices have been attracting a great deal of attention theoretically. Related experimental developments have provided the means to verify many of the theoretical predictions. In addition to ultracold fermionic gases taking the role of electrons in solid state materials [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], also bosons, for example loaded into the $p$ band, exhibit exotic many-body quantum phases due to interaction effects [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. A pioneering experimental work of Ref. [24] successfully realized a $p$-orbital superfluid in a bipartite optical square lattice. In this work, bosons condensed at two quasi-momenta located at the centers of the four boundaries forming the rectangular first Brillouin zone (BZ). The corresponding complex-valued locally chiral $p_{x}\pm ip_{y}$ order parameter breaks time-reversal symmetry. Recently, experimental efforts on $p$-orbital physics have been directed to hexagonal optical lattices [36, 37, 38, 39]. In our recent work [38], we observed the formation of a globally chiral atomic superfluid, induced by interaction-driven time-reversal symmetry breaking in the second Bloch band of a hexagonal boron nitride (BN) optical lattice. The emergent orbital superfluid supports topological Bogoliubov excitations and edge states.

In experiments with bosons in higher Bloch bands with orbital degrees of freedom, time-reversal symmetry and discrete rotation symmetries must be implemented with high precision in order to realize and exploit orbital degeneracies. In this manuscript, we report a versatile high precision scheme to construct a hexagonal BN optical lattice and propose a strategy to precisely optimize the optical lattice geometry. Unlike other schemes [40, 36, 41, 37], we create the hexagonal optical lattice by overlapping two triangular optical lattices, which contribute sublattice sites $\mathcal{A}$ and $\mathcal{B}$ to the resulting BN lattice. Each triangular lattice is generated by intersecting three monochromatic running wave laser beams with out-of-plane linear polarization. The potential depth and spatial position of each sublattice can be adjusted independently in the experiment. We swap the potential depths of the two sublattice wells to load atoms into the desired second band. In this process we added an extra step of reducing the superimposed optical dipole trap to further cool the atoms, which turned out crucial for the realization of long-lived condensates with very low temperature.

Since the energy minima of the second band sensitively depend on the relative positions of the two triangular lattices, small variations of the experimental parameters would cause a significant change in the locations of Bragg resonances in momentum spectra recorded by ballistic time-of-flight (TOF) and subsequent absorption imaging. We thus developed a strategy to optimize the optical lattice geometry. First, we calibrate the positions of Bragg peaks from the momentum spectra, which reflect the corresponding quasi-momenta for the energy minima in the second band. Second, we deduce the current experimental parameters assisted by numerical calculations and tune these parameters to correct the lattice deformation. This strategy has been readily applied in the experiment and we believe that it could be easily generalized to other optical lattice systems.

The remainder of this manuscript is organized as follows: In Sec. 2, we introduce the hexagonal optical lattice potential and analyze its deformation with the variation of experimental parameters. In Sec. 3, we describe the protocols for building and controlling the experimental setup. In Sec. 4, we show the experimental results of the TOF images of the condensates, and discuss our strategy to optimize the optical lattice potential. A summary and an outlook are given in Sec. 5.

Theoretically, we apply three laser beams intersecting at $120^{\circ}$ angle with respect to each other in the $xy$-plane to construct the hexagonal lattice potential. The polarizations of all laser beams are parallel to the $z$-axis. Each laser beam comprises two spectral components with slightly different wavelengths $\lambda_{1}$ and $\lambda_{2}$ around $1064\,$nm. The interference among each set of spectral components with the same frequency generates a triangular lattice. Therefore, the total hexagonal optical lattice potential is a superposition of two triangular lattices with corresponding lattice sites denoted respectively by $\mathcal{A}$ and $\mathcal{B}$ and can be written as $V_{h}(\mathbf{r})=V_{\lambda_{1}}(\mathbf{r})+V_{\lambda_{2}}(\mathbf{r})$ with

	$$V_{\lambda_{1}}(\mathbf{r})=-V_{1}\left\{3+2\sum_{\left<i,j\right>}\cos\left[\left(\mathbf{k}_{i}-\mathbf{k}_{j}\right)\cdot\mathbf{r}+\left(\theta_{i}-\theta_{j}\right)\right]\right\},$$		(1a)
	$$V_{\lambda_{2}}(\mathbf{r})=-V_{2}\left\{3+2\sum_{\left<i,j\right>}\cos\left[\left(\mathbf{k}_{i}-\mathbf{k}_{j}\right)\cdot\mathbf{r}+\left(\theta^{\prime}_{i}-\theta^{\prime}_{j}\right)\right]\right\}.$$		(1b)

Here, $V_{1,2}\geq 0$ as $\lambda_{1,2}$ are far red detuned from the atomic resonance. The summation $\left<i,j\right>$ is limited to $\left<1,2\right>$, $\left<2,3\right>$, $\left<3,1\right>$. The wave vectors are given by $\mathbf{k}_{1}=k_{\mathrm{L}}(-\sqrt{3}/2,1/2)$, $\mathbf{k}_{2}=k_{\mathrm{L}}(\sqrt{3}/2,1/2)$, and $\mathbf{k}_{3}=k_{\mathrm{L}}\left(0,-1\right)$ with $k_{\mathrm{L}}=2\pi/\lambda_{1}$. The phase angles are determined by $\theta_{j}=2\pi\nu_{1}L_{j}/c+\theta_{0}$ and $\theta^{\prime}_{j}=2\pi\nu_{2}L_{j}/c+\theta_{0}^{\prime}$, where $\nu_{1,2}=c/\lambda_{1,2}$ are frequencies and $L_{j}$ denotes the optical path length of the $j$-th beam from the splitting point to the center of the lattice. $c$ denotes the speed of light. $\theta_{0}$ and $\theta_{0}^{\prime}$ include the phase information for laser beams and are irrelevant to the lattice structure.

This optical lattice system has various degrees of freedom to be conveniently adjusted in the experiment. $V_{1,2}$ control the potential well depths of the two triangular sublattices independently. The relative position of the two triangular lattices is described by $\Delta\mathbf{r}$ which is defined via the equation $V_{\lambda_{1}}(\mathbf{r})/V_{1}=V_{\lambda_{2}}(\mathbf{r}-\Delta\mathbf{r})/V_{2}$ and can be explicitly written as

$$\Delta\mathbf{r}=\frac{\sqrt{3}a_{0}}{4\pi}\left(\sqrt{3}\Delta\theta_{12},-\Delta\theta_{12}-2\Delta\theta_{23}\right).$$

(2)

Here, $a_{0}=4\pi/(3\sqrt{3}k_{\mathrm{L}})$ and the relative phase differences $\Delta\theta_{ij}$ are defined as

$$\Delta\theta_{ij}=(\theta_{i}-\theta_{j})-(\theta_{i}^{\prime}-\theta_{j}^{\prime})=\frac{2\pi\Delta\nu}{c}(L_{i}-L_{j})$$

(3)

with $\Delta\nu=\nu_{1}-\nu_{2}$. Note that there are only two independent degrees of freedom as $\Delta\theta_{12}+\Delta\theta_{23}+\Delta\theta_{31}=0$. Thus, $\Delta\mathbf{r}$ can be rewritten as

$$\Delta\mathbf{r}=\frac{\sqrt{3}a_{0}\Delta\nu}{2c}\left(\sqrt{3}L_{1}-\sqrt{3}L_{2},-L_{1}-L_{2}+2L_{3}\right).$$

(4)

When fixing $L_{3}=(L_{1}+L_{2})/2$, we can change $L_{1}-L_{2}$ or $\Delta\nu$ to tune the relative position of the two triangular sublattices along the $x$-direction. If we further choose $(L_{1}-L_{2},L_{2}-L_{3},L_{3}-L_{1})=(-6.04,3.02,3.02)\ \text{cm}$ and $\Delta\nu=3.308$ GHz, $\Delta\mathbf{r}=(-a_{0},0)$ and a regular hexagonal optical lattice is formed. The unit cell in this case is a hexagon as shown in Fig. 1 (a). Adjacent vertices are associated with potential wells denoted by $\mathcal{A}$ and $\mathcal{B}$, respectively. The potential maximum is located at the center of the unit cell. A typical resulting hexagonal optical lattice potential is shown in Fig. 1 (b).

Our experiment starts with a ⁸⁷Rb condensate with approximately $4\times 10^{4}$ atoms in the hyperfine ground state $|F=1,m_{F}=-1\rangle$ without discernable thermal atoms, produced in a pancake optical dipole trap. The dipole trap is formed by two elliptical beams perpendicular to each other at the center of an ultra-high dodecagonal vacuum metal chamber, which is shown in Fig. 2 (a). The trapping frequencies are $\{\omega_{x},\omega_{y},\omega_{z}\}=2\pi\times\{26.4,26.7,70.6\}$Hz. A two-dimensional hexagonal optical lattice is formed by superimposing two sets of triangular lattices generated by three two-color beams.

Experimentally, to create a regular hexagonal lattice we need to fine tune all parameters including $L_{1}$, $L_{2}$, $L_{3}$, and $\Delta\nu$. First, we measure and roughly set the values of $(L_{1}-L_{2},L_{2}-L_{3},L_{3}-L_{1})$ as $(-6,3,3)$ cm. Second, we turn on two out of three lattice beams to form a 1D optical lattice. Loading a Bose-Einstein condensation into the lattice, we calibrate the lattice depth via amplitude modulated spectroscopy and obtain information for the lattice center difference when we turn on all three lattice beams. As an example, we consider the $i$-th and $j$-th lattice beams with four wave vectors $(\mathbf{k}_{i},\mathbf{k}_{j},\mathbf{k}_{i}^{\prime},\mathbf{k}_{j}^{\prime})$. In the beginning, we tune the power of each spectral component of each beam to make sure that $V_{1}=V_{2}$ (denoted as $V_{0}$). Their combination gives rise to a 1D lattice potential, which is given by

$\displaystyle V_{ij}^{\rm(1D)}(\mathbf{r})=-4V_{0}-V_{ij}\cos\left\{\left(\mathbf{k}_{i}-\mathbf{k}_{j}\right)\cdot\mathbf{r}+\frac{1}{2}\left[(\theta_{i}-\theta_{j})+(\theta_{i}^{\prime}-\theta_{j}^{\prime})\right]\right\}$

(5)

where $V_{ij}=4V_{0}\cdot\cos\left(\frac{\pi\Delta v}{c}\left(L_{i}-L_{j}\right)\right)$.

We then change the frequency difference among two spectral components in a range of $(0,10)$ GHz and measure $(|V_{12}|,|V_{23}|,|V_{31}|)$ accordingly. Fig. 2(b) shows our final results after we carefully set $L_{1}$, $L_{2}$, and $L_{3}$. Note that we use the fact that $V_{23}=V_{31}$ holds for arbitrary values of $\Delta\nu$ when $L_{2}-L_{3}=L_{3}-L_{1}$ to fine tune $L_{3}$. In Fig. 2(b), we fit the experimental data by applying the formula

$\displaystyle|V_{ij}(\Delta\nu)|=a_{ij}\left|\,\cos\frac{2\pi\Delta\nu}{b_{ij}}\,\right|,$

(6)

where $a_{ij}$ and $b_{ij}$ are the fitting parameters. The fitting results are $(a_{12}=7.347(8)\ E_{R},b_{12}=9.909(6)\,\rm GHz)$, $(a_{23}=7.38(2)\ E_{R},b_{23}=19.87(2)\,\rm GHz)$, $(a_{31}=7.38(1)\ E_{R},b_{31}=19.84(2)\,\rm GHz)$ in the confidence level of 95$\%$ for curves. Based on these parameters, we obtain more precision values for optical path differences as $(L_{1}-L_{2},L_{2}-L_{3},L_{3}-L_{1})=(-6.04,3.02,3.02)$ cm. When the frequency difference is set at $\Delta\nu=3.308\,\rm GHz$ or $6.607\,\rm GHz$ as marked by two circles in Fig. 2(b), a regular hexagonal is created.

In conclusion, we apply an effective high-precision scheme to construct a highly tunable precisely hexagonal optical lattice in the experiment. Furthermore, we propose and implement strategies to optimize the lattice geometry by carefully determining the positions of Bragg peaks from momentum distributions of the condensate in the second Bloch band. All experimental results are in good agreement with theoretical calculations. Our work paves the way for investigating many-body physics in the hexagonal optical lattice, especially for novel higher orbital physics, which sensitively depends on the lattice rotational symmetry. The schemes of constructing and optimizing the optical lattice can potentially be extended to other non-square lattices.

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Acknowledgments This work is supported by the Key-Area Research and Development Program of Guangdong Province (Grant No. 2019B030330001), the National Key R&D Program of China (Grant No. 2018YFA0307200), NSFC (Grant No. U1801661), and a fund from Guangdong province (Grant No. 2019ZT08X324). X.-Q. W. acknowledges support from the China Postdoctoral Science Foundation (No. 2021M691444). A.H. acknowledges support by Cluster of Excellence CUI: Advanced Imaging of Matter of the Deutsche Forschungsgemeinschaft (DFG) - EXC 2056 - project ID 390715994.

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Disclosures The authors declare no conflicts of interest.

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Data Availability Statement Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.