Matter waves in two-dimensional arbitrary atomic crystals

We present a general scheme to realize a cold-atom quantum simulator of bidimensional atomic crystals. Our model is based on the use of two independently trapped atomic species: the first one, subject to a strong in-plane confinement, constitutes a two-dimensional matter wave which interacts only with atoms of the second species, deeply trapped around the nodes of a two-dimensional optical lattice. By introducing a general analytic approach we show that the system Green function can be exactly determined, allowing for the investigation of the matter-wave transport properties. We propose some illustrative applications to both Bravais (square, triangular) and non-Bravais (graphene, kagomé) lattices, studying both ideal periodic systems and experimental-size and disordered ones. Some remarkable spectral properties of these atomic artificial lattices are pointed out, such as the emergence of single and multiple gaps, flat bands, and Dirac cones. All these features can be manipulated via the interspecies interaction, which proves to be widely tunable due to the interplay between scattering length and confinements.

Figure 1

Schematic of our model for the realization of 2D atomic crystals. Two atomic species, namely, $A$ (depicted in blue) and $B$ (depicted in red), are strongly confined on a plane. Making use of a species-selective optical lattice [10, 11], $B$ atoms are arranged in a 2D lattice of point-like scatterers of arbitrary geometry (a square one in this example), while $A$ atoms form a matter wave which propagates through the artificial crystal.

Figure 2

Schematic of the two-body scattering process between a 2D-trapped $A$ atom and a 0D-trapped $B$ atom. The scattering process in the actual system (top) is ruled by the 3D $A-B$ scattering length $a_{3\mathrm{D}}$ of the contact interaction, as well as by the trapping frequencies and the mass ratio of the two atomic species. The system is equivalent to a strictly bidimensional one (bottom) in which the $B$ atom constitutes a fixed point-like scatterer and the scattering process is ruled only by an effective 2D scattering length $a_{2\mathrm{D}}^{\mathrm{eff}}$ [20].

Figure 3

Left: Schematic in real space of a square lattice of primitive vectors ${\mathbf{a}}_1=a(1,0)$ and ${\mathbf{a}}_2=a(0,1)$. The shaded area represents the unit cell. Right: Reciprocal lattice in $k$ space corresponding to the square lattice. Consequently the reciprocal primitive vectors are ${\mathbf{b}}_1=\frac{2\pi }{a}(1,0)$ and ${\mathbf{b}}_2=\frac{2\pi }{a}(0,1)$. The first Brillouin zone (FBZ) is shaded. The high-symmetry points $\Gamma =(0,0)$, $\mathbf{X}=\frac{\pi }{a}(1,0)$, and $\mathbf{M}=\frac{\pi }{a}(1,1)$ are highlighted and the $\Gamma -\mathbf{X}-\mathbf{M}-\Gamma$ path [dashed (red) line] constitutes an irreducible symmetry path.

Figure 4

Left: Real-space representation of a triangular lattice of primitive vectors ${\mathbf{a}}_1=\frac{3a}{2}(1,1/\sqrt{3})$ and ${\mathbf{a}}_2=\frac{3a}{2}(1,-1/\sqrt{3})$. The shaded region represents the Wigner-Seitz unit cell. Right: Reciprocal lattice for the triangular one, ${\mathbf{b}}_1=\frac{2\pi }{3a}(1,\sqrt{3})$ and ${\mathbf{b}}_2=\frac{2\pi }{3a}(1,-\sqrt{3})$ being the reciprocal primitive vectors. The first Brillouin zone (FBZ) is shaded and the high-symmetry points $\Gamma =(0,0)$, $\mathbf{M}=\frac{2\pi }{3a}(1,0)$, $\mathbf{K}=\frac{2\pi }{3a}(1,1/\sqrt{3})$, and ${\mathbf{K}}^{\prime }=\frac{2\pi }{3a}(1,-1/\sqrt{3})$ are highlighted. The $\Gamma -\mathbf{M}-\mathbf{K}-\Gamma$ path [dashed (red) line] is a high-symmetry one.

Figure 5

For a square lattice (cf. Fig. 3), behavior of $f(\mathbf{q},E)$ as a function of $E$ for different values of $\mathbf{q}$ [namely, $\frac{\pi }{a}(0.8,0.5)$, $\frac{\pi }{a}(0.9,0.5)$, and $\frac{\pi }{a}(1,0.5)$, from top to bottom]. Dashed vertical (gray) lines represent the lowest values of $E_{\mathrm{free}}$ for the selected $\mathbf{q}$. In the sketches at the right of each panel a point indicates the corresponding $\mathbf{q}$ inside the $\Gamma -\mathbf{X}-\mathbf{M}-\Gamma$ symmetry path. Here $\varepsilon ={\hslash }^2/m{a}^2$.

Figure 6

For the artificial atomic square lattice, the band structure (left) and density of states (right) evaluated at $\alpha =-0.75$. Left: The four lowest energy bands are evaluated along the irreducible symmetry path $\Gamma -\mathbf{X}-\mathbf{M}-\Gamma$ within the first Brilouin zone (see inset). Dashed (gray) lines correspond to the energy spectrum of a free matter wave. Right: Density of states (DOS) obtained by evaluating the energies of the same bands in $N_s\simeq 6100$ points sampled inside the path. For each bin of the histogram (of width $\delta E=0.1\varepsilon$, $\varepsilon ={\hslash }^2/m{a}^2$) the band-normalized DOS is evaluated according to Eq. (19).

Figure 7

Square lattice: band structure for different values of the interaction parameter $\alpha$. Spectra are evaluated along the same path as shown in Fig. 6. Dashed (gray) lines correspond to the energy spectrum of a free matter wave. Energies are normalized on $\varepsilon ={\hslash }^2/m{a}^2$.

Figure 8

Finite-sized square lattice: DOS per scatterer in the plane $[\alpha ,E/\varepsilon ]$ for a system of $N=2551$ scatterers arranged in a square lattice inside a disk of radius $R=26a$. Energies are discretized with a step of $0.01\varepsilon$ ($\varepsilon ={\hslash }^2/m{a}^2$). For a given $E<0$ all $N$ solutions of Eqs. (9) are selected. For $E>0$ the sampled values are used as starting points to find $N$ complex poles of $G$ according to Eq. (20). Only quasi-Bloch bulk states are selected, as explained in the text. The color map is applied to ${log}_{10}\left(\frac{N_p}{N}\frac{\varepsilon }{\delta \alpha \delta E}\right)$, where $N_p$ is the number of selected poles of $G$ within a rectangular bin of area $\delta \alpha \delta E$ ($\delta \alpha =0.02$ and $\delta E=0.05\varepsilon$). White circles indicate the positions of band boundaries as expected from the analysis of an infinite system (Figs. 6 and 7). Analogously the white plus symbol marks the expected contact point between the lowest and the first excited band, corresponding to the gap closure.

Figure 9

Square lattice of finite size: normalized densities of lifetimes $\tau$ for $E/\varepsilon =6.5\pm 0.5$ and $\alpha =-0.75\pm 0.1$ ($\varepsilon ={\hslash }^2/m{a}^2$). Different panels correspond to different radii $R$ of the scattering region. Dot-dashed vertical (red) lines in each panel indicate $R/{\overline{v}}_g$, where ${\overline{v}}_g$ is the average group velocity in the selected region of the $[\alpha ,E]$ plane.

Figure 10

For the same finite-size square lattice considered in Fig. 8: normalized density of bandwidths $\Gamma$ for all positive-energy poles of the Green function ($\varepsilon ={\hslash }^2/m{a}^2$). The dot-dashed vertical (blue) line at $\Gamma ={\Gamma }_{\max }$ marks the change in behavior of the density ${\rho }_{\Gamma }$. Solutions with $\Gamma >{\Gamma }_{\max }$ have been rejected because they are incompatible with the behavior of a quasi-Bloch bulk state. Here we found ${\Gamma }_{\max }\simeq 0.7\varepsilon /\hslash$ (i.e., ${\Gamma }_{\max }\simeq 2\mathrm{kHz}$ for a matter wave of ${}^{87}\mathrm{Rb}$ atoms in an AASL with $a=500\mathrm{nm}$).

Figure 11

Disordered square lattice: normalized negative-energy DOS at $\alpha =-0.75$ for an AASL of radius $R=32a$, corresponding to 3209 available lattice sites. Different panels refer to different percentages of randomly occupied sites: (a) 100%, (b) 95%, (c) 20%, and (d) 10%. Histograms are obtained with a bin size $\delta E=0.05\varepsilon$ in (a) and (b) and $\delta E=0.025\varepsilon$ in (c) and (d). Superimposed dotted histograms in (a) and (b) represent the DOS for an infinite periodic system. Vertical lines in (c) and (d) indicate the energies of few-body bound states: $AB$ dimer (solid black line), $A{B}_2$ trimer with $B$ atoms separated by $a$ [dashed (blue) line] and $a\sqrt{2}$ [dot-dashed (red) line]. Here $\varepsilon ={\hslash }^2/m{a}^2$.

Figure 12

Artificial atomic triangular lattice: band structure (left) and DOS (right) for $\alpha =-0.6$. Left: Behavior of the four lowest energy bands along the $\Gamma -\mathbf{M}-\mathbf{K}-\Gamma$ path (see inset at right). Dashed (gray) lines indicate the dispersion relation for a free matter wave. Right: DOS, as defined in Eq. (19), obtained by evaluating the energy of the bands in $N_s=7600$ points sampled inside the aforementioned symmetry path. The bin size of the histogram is $\delta E=0.05\varepsilon$ ($\varepsilon ={\hslash }^2/m{a}^2$).

Figure 13

Triangular lattice: comparison of the band structure for different values of the interaction parameter $\alpha$. The spectrum evaluation path is the same as for Fig. 12. Dashed (gray) lines correspond to the energy spectrum of a free matter wave. Energies are normalized on $\varepsilon ={\hslash }^2/m{a}^2$.

Figure 14

Triangular lattice of finite size: DOS per scatterer in the plane $[\alpha ,E/\varepsilon ]$ for a system of $N=2125$ scatterers arranged in a triangular lattice inside a disk of radius $R=42a$. The same method as for Fig. 8 was used, but discretizing $E$ in steps of $0.005\varepsilon$ ($\varepsilon ={\hslash }^2/m{a}^2$). Positive-energy quasi-Bloch bulk states were selected imposing $\Gamma <{\Gamma }_{\max }\simeq 0.3\varepsilon /\hslash$ (i.e., ${\Gamma }_{\max }\simeq 880$ Hz for a matter wave of ${}^{87}\mathrm{Rb}$ atoms in an AATL with $a=500\mathrm{nm}$). The color map is applied to the quantity ${log}_{10}\left(\frac{N_p}{N}\frac{\varepsilon }{\delta \alpha \delta E}\right)$, where $N_p$ is the number of selected poles of $G$ within a rectangular bin of area $\delta \alpha \delta E$ ($\delta \alpha =0.02$ and $\delta E=0.025\varepsilon$). White circles indicate the expected positions of the band boundaries as evaluated for the infinite system (Figs. 12 and 13). Analogously, the white plus symbol marks the expected contact point between the lowest and the first excited band.

Figure 15

(a) Representation of an $M=2$ non-Bravais lattice. It can be seen as a triangular lattice (Fig. 4) with two atoms per unit cell (shaded hexagon) whose relative position is ${\mathbf{t}}_{12}$. Equivalently, it can be obtained as the superposition of two interpenetrating triangular lattices (distinguished by colors) displaced by ${\mathbf{t}}_{12}$ with respect to each other. The generating vectors ${\mathbf{a}}_{1,2}=\frac{3a}{2}(1,\pm 1/\sqrt{3})$ are also indicated. (b) Representation of the triangular lattice as an example of a basic Bravais lattice, i.e., with one atom per unit cell. The distance between nearest atoms is $a\sqrt{3}$. (c) Representation of the hexagonal lattice of graphene, a two-atom non-Bravais lattice based on the triangular one, obtained for ${\mathbf{t}}_{12}=({\mathbf{a}}_1+{\mathbf{a}}_2)/3=(a,0)$. With these definitions, the side of the hexagons has length $a$. (d) Representation of the kagomé lattice, a three-atom non-Bravais lattice based on the triangular one, obtained for ${\mathbf{t}}_{12}={\mathbf{a}}_1/2$ and ${\mathbf{t}}_{13}={\mathbf{a}}_2/2$. The nearest-neighbor distance is $a\sqrt{3}/2$.

Figure 16

Atomic artificial graphene. Top: Behavior of $t_{+}^o(\mathbf{q},E)$ [solid (purple) line] and $t_{-}^o(\mathbf{q},E)$ [dashed (green) line] as a function of $E$ for $\mathbf{q}=\frac{2\pi }{3a}(0.8,0.3)$. Dashed vertical (gray) lines indicate the lowest values of $E_{\mathrm{free}}$ for the selected $\mathbf{q}$. Dot-dashed (blue) curves show the behavior of $f(\mathbf{q},E)$. Inset: Position of $\mathbf{q}$ within the $\Gamma -\mathbf{M}-\mathbf{K}-\Gamma$ symmetry path (Fig. 4). Bottom: Behavior of $\left|\phi \right(\mathbf{q},E\left)\right|$ for the same $\mathbf{q}$. Here $\varepsilon ={\hslash }^2/m{a}^2$.

Figure 17

Honeycomb lattice: band structure (left) and DOS (right) for AAG at $\alpha =-0.6$. Left: Behavior of the five lowest energy bands along the $\Gamma -\mathbf{M}-\mathbf{K}-\Gamma$ symmetry path (see inset at right). Dashed (gray) curves show the band structure for the corresponding triangular lattice. Right: DOS [defined in Eq. (19)] obtained by sampling the band energy in $N_s=7600$ points within the symmetry path. The histogram bin size is $\delta E=0.05\varepsilon$ ($\varepsilon ={\hslash }^2/m{a}^2$).

Figure 18

Atomic artificial graphene: comparison of the band structures for different values of $\alpha$ evaluated along the $\Gamma -\mathbf{M}-\mathbf{K}-\Gamma$ symmetry path (see inset in Fig. 17). For $\alpha \simeq -1.631$ the third, isolated band is topologically flat. Dashed (gray) curves show the band structure for the corresponding triangular lattices. The unit of energy is $\varepsilon ={\hslash }^2/m{a}^2$.

Figure 19

Kagomé lattice: comparison of the band structures of the artificial lattice for different values of $\alpha$ evaluated along the $\Gamma -\mathbf{M}-\mathbf{K}-\Gamma$ symmetry path within the FBZ (see inset). The two lowest bands touch in $\mathbf{q}=\mathbf{K},{\mathbf{K}}^{\prime }$, forming a Dirac cone. For $\alpha \simeq -1.98$ the fourth, isolated band is topologically flat. For $\alpha <-1$ the three lowest bands lie entirely at $E<6.5\varepsilon$ ($\varepsilon ={\hslash }^2/m{a}^2$) and are not shown in the plot. Dashed (gray) lines correspond to the energy spectrum of a free matter wave.

Figure 20

Finite-sized kagomé lattice: DOS per scatterer in the plane $[\alpha ,E/\varepsilon ]$ for a system of $N=2083$ scatterers arranged in a kagomé lattice inside a disk of radius $R=24a$. Results were obtained with the same method as described for Fig. 8, discretizing the energies with a step of $0.005\varepsilon$ ($\varepsilon ={\hslash }^2/m{a}^2$). Quasi-Bloch bulk states were selected choosing ${\Gamma }_{\max }\simeq 0.5\varepsilon /\hslash$ (i.e., ${\Gamma }_{\max }\simeq 1460$ Hz for a matter wave of ${}^{87}\mathrm{Rb}$ atoms for $a=500\mathrm{nm}$). The color map is applied to the quantity ${log}_{10}\left(\frac{N_p}{N}\frac{\varepsilon }{\delta \alpha \delta E}\right)$, where $N_p$ is the number of selected poles of $G$ within a rectangular bin of area $\delta \alpha \delta E$ ($\delta \alpha =0.02$ and $\delta E=0.025\varepsilon$). White $\mathsf{\text{X}}$, circles, and arrows indicate, respectively, the positions of Dirac cones, gap boundaries, and the isolated flat band as expected from the analysis of an infinite system (Fig. 19).