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.

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• Coherent control
• Quantum Hall effect
• Quantum simulation
• Topological phases of matter
• Rydberg atoms & molecules
• Optical tweezers