Semisynthetic zigzag optical lattice for ultracold bosons

We propose a cold-atom realization of a zigzag ladder. The two legs of the ladder correspond to a “synthetic” dimension given by two internal (spin) states of the atoms, so that tunneling between them can be realized as a laser-assisted process. The zigzag geometry is achieved by employing a spin-dependent optical lattice with the site position depending on the internal atomic state, i.e., on the ladder's leg. The lattice offers a possibility to tune the single-particle dispersion from a double-well to a single-minimum configuration. In contrast to previously considered semisynthetic lattices with a square geometry, the tunneling in the synthetic dimension is accompanied by spatial displacements of atoms. Therefore, the atom-atom interactions are nonlocal and act along the diagonal (semisynthetic) direction. We investigate the ground-state properties of the system for the case of strongly interacting bosons. In particular, we find that the interplay between the frustration induced by the magnetic field and the interactions gives rise to an interesting gapped phase at fractional filling factors corresponding to one particle per magnetic unit cell.

Figure 1

(a) Semisynthetic zigzag lattice corresponding to the tight-binding Hamiltonian (6). The lattice is affected by a nonstaggered flux $\gamma =a\tilde{\kappa }/2=\pi \tilde{\kappa }/\kappa$ over triangular plaquettes. (b) Tight-binding parameters. Left: Horizontal nearest-neighbor ($t$) and next-nearest-neighbor ($t^{\left(2\right)}$) hopping parameters together with diagonal tunneling strength $t_D$ as a function of the scaled lattice depth $V_0/E_R$ for $\mathrm{\Omega }=0.2020E_R$. Right: Normalized interactions $U_1$ and $U_2$ as functions of the scaled lattice depth.

Figure 2

(a) Schematic layout of proposed experimental setup. A bias magnetic field ${\mathbf{B}}_0=B_0{\mathbf{e}}_x$ Zeeman splits the hyperfine spin states of ${}^{87}\mathrm{Rb}$ atoms. A counterpropagating pair of ${\sigma }_{+}$ polarized laser beams (shown in red) with ${\lambda }_L=4\pi /\kappa \approx 789\mathrm{nm}$ form a spin-dependent lattice with opposite signs for atoms in the $m_F=-1$ and $m_F=0$ states [8, 38, 39]. This traps the $|F=1,m_F=-1,0\rangle$ states on lattice sites shifted by half the lattice constant $a/2=\pi /\kappa$ providing a semisynthetic zigzag lattice. Two horizontally polarized lasers (shown in red) at ${\lambda }_R=2\pi /\tilde{\kappa }\approx 790\mathrm{nm}$ resonantly couple the two spin states producing the flux $\gamma =\tilde{\kappa }a/2=\pi /2$ tuned by taking the angle $\alpha \approx {60}^{\circ }$ between the laser beams. (b) Level diagram. The ${\sigma }_{+}$ polarized lattice laser beams (dashed red arrows) shift individual $m_F$ states, but do not drive transitions. The strength of the state-dependent contribution to this shift is maximized when the beams propagate parallel to ${\mathbf{B}}_0$. The Raman lasers (solid green arrows) are tuned to be in resonance with $m_F=-1\to 0$ transition, but detuned from the $m_F=0\to +1$ transition. Note that the Zeeman splitting of the ground-state $F=1$ hyperfine manifold is shown on a greatly exaggerated scale.

Figure 3

Exactly calculated dispersion curves for $V_0=5E_R, \kappa =2\tilde{\kappa }$ ($\gamma =\pi /2$), and various strengths of the spin-flip coupling: (a) $\mathrm{\Omega }=0.05$, (b) $\mathrm{\Omega }=0.2020$, (c) $\mathrm{\Omega }=0.4039E_R$ (the critical value), and (d) $\mathrm{\Omega }=0.8E_R$. This corresponds to: (a) $t_D=0.25t$, (b) $t_D=t$, (c) $t_D=2t$ (quartic dispersion at $k=0$), and (d) $t_D=4t$.

Figure 4

Bloch oscillations for a Gaussian wave packet in a tilted zigzag lattice. Panel (a) corresponds to the band structure presented in Fig. 2 of the main text. In panel (b) the legs of the ladder legs are additionally biased by introducing a spin-dependent onsite energy shift $\pm 0.3t{\sigma }_z$, and the band gaps opened close to the Brillouin zone boundary lead to the Landau-Zener tunneling between the two bands.

Figure 5

Many-body phase diagram of the zigzag lattice for a set of flux values $\gamma =\{0.4\pi ,0.5\pi ,0.55\pi ,\frac{2}{3}\pi ,\frac{3}{4}\pi \}$. Top row: the scaled charge gap ${\mathrm{\Delta }}_c/E_R$ plotted as a function of the number of particles $N$ in the lattice of size $L=60$ and the ratio of the hopping strengths $t_D/t$. Areas corresponding to enhanced charge gaps close to $t_D\approx t$ and the filling factor $N/L=\gamma /2\pi$ are conspicuous and are marked with white ovals. Bottom row: the expectation of the site occupation $\langle n_{\ell }\rangle$ versus the site number $\ell$ calculated at the phase-diagram points inside the white ovals.

Figure 6

Spin polarization $\mathcal{P}$ as a function of the nearest-neighbor repulsion strength calculated for $N=30$ particles on a zigzag lattice of $L=59$ sites without the artificial flux. When diagonal transitions dominate over the horizontal transitions, $t_D\ge t$, the shown dependence of the polarization $\mathcal{P}$ becomes insensitive to the relative magnitude of $t$. The thick line is a superposition of many dependencies with the ratio of the hopping parameters $1\le t_D/t\le 20$.