Quantum Hall states for Rydberg arrays with laser-assisted dipole-dipole interactions

Rydberg atoms with dipole-dipole interactions provide intriguing platforms to explore exotic quantum many-body physics. Here we propose a mechanism dubbed laser-assisted dipole-dipole interactions to realize synthetic magnetic field for Rydberg atoms in a two-dimensional array configuration, which gives rise to the exotic bosonic topological states. In the presence of an external effective Zeeman splitting gradient, the dipole-dipole interaction between neighboring Rydberg atoms along the gradient direction is suppressed but can be assisted when Raman lights are applied to compensate the energy difference. With this scheme we generate a controllable uniform magnetic field for the complex spin-exchange coupling model, which can be mapped to hard-core bosons coupling to an external synthetic magnetic field. The highly tunable flat Chern bands of the hard-core bosons are then obtained and, moreover, the bosonic fractional quantum Hall states can be achieved with experimental feasibility. This Letter opens up an intriguing avenue for the realization of the highly sought-after bosonic topological orders using Rydberg atoms.

Figure 1

Sketch of the proposal. (a) Rydberg atoms trapped in optical tweezers to form a 2D array. An energy shift with gradient in $y$ direction modifies the energy difference between the two Rydberg states (effective Zeeman splitting for pseudospin). (b) The Rydberg dipole-dipole interaction can induce spin-exchange coupling between adjacent sites along the $x$ direction. (c) In the $y$ direction, the spin-exchange coupling is suppressed by the effective Zeeman energy offset. A two-photon Raman process compensates this energy offset and drives the laser-assisted spin-exchange couplings. (d) The laser-assisted exchange couplings have a spatially dependent phase, generating a synthetic magnetic flux in the 2D lattice.

Figure 2

Numerical simulation for the laser-assisted exchange couplings and the synthetic magnetic flux under ${\varphi }_y=\pi$. (a) Raman coupling driven Rabi oscillation on a two-site system, which has an energy offset $\mathrm{\Delta }$ compensated by the Raman potential. A boson is initialized at site 1 and evolves afterwards. Take $\eta =2|{\mathrm{\Omega }}_1{\mathrm{\Omega }}_2/\left(\mathrm{\Delta }{\mathrm{\Delta }}_i\right)|=0.5$ and $\mathrm{\Delta }=5J_y^0=10J_y$. The slow oscillation is driven by the Raman coupling, whereas the fast oscillations of frequency $\mathrm{\Delta }$ correspond to the off-resonant bare transition. (b) The amplitude of $J_y^{\mathrm{eff}}$ matches well with the perturbative result $|J_y^{\mathrm{eff}}|=\eta J_y^0$ when $\eta$ and $J_y^0/\mathrm{\Delta }$ are small. When $\eta$ is large, $|J_y^{\mathrm{eff}}|$ deviates clearly from the perturbation results. (c) The phase of the exchange-couplings $J_y, J_{d_1}$, and $J_{d_2}$ (not shown in figure). The numerical results show good coincidence with perturbation results.

Figure 3

The quantum Hall bands modulated by the diagonal hoppings $J_{d_1\left(d_2\right)}$ when $J_x^0=J_y^0$. (a) The deformed Hofstadter butterfly is generated at ${\varphi }_y=0.8\pi$. (b) and (c) The Chern bands for different ${\varphi }_y$'s and $\mathrm{\Phi }$'s. (d) The flatness ratio $E_{\mathrm{gap}}/W$ versus ${\varphi }_y$. The red solid lines show the flatness ratio for the realized model in comparison with the case of setting $J_d=0$ by hand (blue dashed line corresponding). The maximal flatness is obtained at ${\varphi }_y={\varphi }_y^o$.

Figure 4

The bosonic $1/2$-fractional quantum Hall state. The low-energy spectra $E_n-E_0$ versus the system size $1/L$ with $L=L_x{L}_y$. The results show the twofold degeneracy of the many-body ground states, which have a finite gap separating from the excited states. Other parameters are taken $J_x=J_y=J$.