Spectra and ground states of one- and two-dimensional laser-driven lattices of ultracold Rydberg atoms

We investigate static properties of laser-driven ultracold Rydberg atoms confined to one- and two-dimensional uniform lattices in the limit of vanishing laser coupling. The spectral structure of square lattices is compared to those of linear chains, and similarities as well as differences are pointed out. Furthermore, we employ a method based on elements of graph theory to numerically determine the laser-detuning-dependent ground states of various lattice geometries. Ground states for chains as well as square and rectangular lattices are provided and are discussed.

Figure 1

Symmetry operations for (a) the chain, (b) the square, and (c) the rectangular lattices.

Figure 2

(a) Spectrum of the chain with 16 sites. (b) and (c) Spectrum of the $4\times 4$ square lattice where (b) accounts only for next- and diagonal-neighbor interactions. For the square lattice, the pattern in the interaction energy occurs on a finer grid compared to the linear chain: The spectrum shows large energy gaps of $\frac{1}{8}{V}_1$ whenever $\Delta /V_1$ is an integral multiple of $\frac{1}{8}$. Due to long-range interactions that lift the degeneracies of energy levels, this pattern is hardly visible for higher energies in the square lattice, cf. upper half of (c). The axes of the subfigures have different scales to ensure the visibility of the structure.

Figure 3

Spectrum of the symmetric subspace of Hamiltonian (1) for the square lattice with $3\times 3$ sites with a laser coupling of $\Omega =0.05V_1$. Red dashed lines: The lower half of the plot shows the projection probabilities of the ground state on a sample of (symmetrized) basis states.

Figure 4

Illustration of the MECs or ground states of the linear chains with (a) $N=15$ and (b) $N=16$. Excitations are red(gray), ground-state atoms are in white. Note, these states are usually degenerate for symmetry reasons. (c) and (d) show a detail of the spectrum where the ground state is indicated by a red dashed line.

Figure 5

Illustration of the MECs of the square with $N=9$. Excitations are red(gray), ground-state atoms are in white. Note these states are usually degenerate for symmetry reasons. Parentheses indicate MECs, which do not appear in the ground-state sequence. These have not been found by max-cut but by a brute-force method.

Figure 6

Same as in Fig. 5 but for $N=16$.

Figure 7

Same as in Fig. 5 but for $N=25$.

Figure 8

Same as in Fig. 5 but for $N=36$.

Figure 9

Same as in Fig. 5 but for $N=49$. The MECs for $N_e=7,17,18,19,22,26,27,43$ are not shown since max-cut did not yield them as part of the ground-state sequence up to a resolution in $\Delta /V_1$ of ${10}^{-8}$. Due to the computational complexity, we refrained from their determination by brute-force methods as we did for smaller lattices.

Figure 10

Same as in Fig. 5 but for a rectangle $N=3\times 7$.

Figure 11

Same as in Fig. 5 but for a rectangle $N=3\times 8$.