Fig. 1. Energy level of two-atoms system excited by one laser.

Fig. 2. Schematic of binary Rydberg energy with detuning: (a) The energy of a pair of atoms with and ; (b) the energy of a pair of atoms with different detuning and potentials^[19].

Fig. 3. Spatially ordered components of the many-body states^[3]: (a) Directly imaging result; (b) accumulative result of many measurements; (c) predicted result.

Fig. 4. (a) Tensor network form of the partition function for 1D Ising model; (b) tensor element for the partition function of 2D Ising model; (c) tensor network form of the partition function for 2D Ising model^[26]

Fig. 5. Relationship between the specific heat and the reciprocal of the temperature^[27].

Fig. 6. Parameters of single-photon source: (a) as a function of effective principle quantum number^[30]. Coincidence count as a function of time decay is showed in the inset^[30]; (b) normalized coincidence count as a function of time decay using quantum dots^[28]; (c) normalized coincidence count of Hong-Ou-Mandel interference as a function of time decay with parallel and cross polarization respectively using quantum dots^[28].

Fig. 7. Properties of quantum storage with different number of input photons N_in^[47]: (a) Storage efficiency as a function of storage time; (b) storage efficiency as a function of Rydberg states.

Fig. 8. Schematic of energy levels combined exciting state with ground state^[4]: (a) Procedure of writing; (b) storage in the ground state; (c) procedure of read.

Fig. 9. Scheme of imaging process and simulated results^[6]: (a) Scheme of single-atom imaging process; (b) absorption of probe light without control light; (c) absorption of probe light with control light.

Fig. 10. (a) Number of retrial gate photons as a function of number of stored gate photons with different number of source photons ^[7]; (b) contrast of optimal efficiency of subtraction^[7].

Fig. 11. phase diagram^[10] and self-organized behaviors^[64]: (a) Phase diagram of density of Rydberg atom; (b) EIT phase diagram without control light; (c) evolution in the self-organized process; (d) regulation of self-organized stationary states.

Fig. 12. Quantum simulation in two dimensions^[63]: (a) Collective Rabi oscillation with different number of atoms; (c) Rydberg fraction of the systems with 20 atoms; (d) Rydberg fraction of the systems with 28 atoms.

Fig. 13. Many-atom quantum simulation in one dimension^[9]: (a) Predicted results of evolution with different interaction; (b) experimental results of evolution with different interaction; (c) ground-state probability as a function of system size; (d) number of states with identical number of occurrences.