Dense atom ensembles with Rydberg excitations display intriguing collective effects mediated by their strong, long-range dipole–dipole interactions. These collective effects, often modeled using Rydberg superatoms, have gained significant attention across various fields due to their potential applications in quantum information processing and quantum optics. In this review article, we delve into the theoretical foundations of Rydberg interactions and explore experimental techniques for their manipulation and detection. We also discuss the latest advancements in harnessing Rydberg collective effects for quantum computation and optical quantum technologies. By synthesizing insights from theoretical studies and experimental demonstrations, we aim to provide a comprehensive overview of this rapidly evolving field and its potential impact on the future of quantum technologies.

We consider a minimal model to describe the quantum phases of ultracold dipolar bosons in two-dimensional square optical lattices. The model is a variation of the extended Bose-Hubbard model and apt to study the quantum phases arising from the variation in the tilt angle ${\displaystyle \theta }$ of the dipolar bosons. At low tilt angles, ${\displaystyle 0^{\circ }⩽\theta \lesssim {25}^{\circ }}$, the ground states of the system are phases with checkerboard order, which can be either checkerboard supersolids or checkerboard density waves. For high tilt angles, ${\displaystyle {35}^{\circ }\lesssim \theta \lesssim {55}^{\circ }}$, phases with striped order of the supersolid or density wave are preferred. In the intermediate domain, ${\displaystyle {25}^{\circ }\lesssim \theta \lesssim {35}^{\circ }}$, an emulsion or superfluid phase intervenes the transition between the checkerboard and the striped phases. The attractive interaction dominates at ${\displaystyle \theta \gtrsim {55}^{\circ }}$, which renders the system unstable, and there is a density collapse. For our studies we use Gutzwiller mean-field theory to obtain the quantum phases and the phase boundaries. In addition, we calculate the phase boundaries between an incompressible and a compressible phase of the system by considering second-order perturbation analysis of the mean-field theory. The analytical results, where applicable, are in excellent agreement with the numerical results. In our study, the incompressible phases have an average occupancy per site ${\displaystyle \rho \le 1}$, but the compressible phases can have ${\displaystyle \rho >1}$.

We study the dynamics of bosonic atoms on a two-dimensional square lattice, where atomic interactions are long ranged with either a box or soft-core shape. The latter can be realized through laser dressing ground-state atoms to electronically excited Rydberg states. When the range of interactions is equal or larger than the lattice constant, the system is governed by an extended Bose-Hubbard model. We propose a quench process by varying the atomic hopping linearly across phase boundaries of the Mott insulator-supersolid and supersolid-superfluid phases. Starting from a Mott insulating state, the dynamical evolution of the superfluid order parameter exhibits a universal behavior at the early stage, largely independent of interactions. The dynamical evolution is significantly altered by strong, long-range interactions at later times. Particularly, we demonstrate that density wave excitation is important when the quench rate is small. Moreover, we show that the quench dynamics can be analyzed through time-of-flight images, i.e., measuring the momentum distribution and noise correlations.

We analyze the physics of self-bound droplets in a doubly dipolar Bose-Einstein condensate composed by particles with both electric and magnetic dipole moments. Using the particularly relevant case of dysprosium, we show that the anisotropy of the doubly dipolar interaction potential is highly versatile and nontrivial, depending critically on the relative orientation and strength between the two dipole moments. This opens novel possibilities for exploring intriguing quantum many-body physics. Interestingly, by varying the angle between the two dipoles we find a dimensional crossover from quasi-one-dimensional to quasi-two-dimensional self-bound droplets. This opens a so far unique scenario in condensate physics, in which a dimensional crossover is solely driven by interactions in the absence of any confinement.

Gases of doubly dipolar particles, with both magnetic and electric dipole moments, offer intriguing novel possibilities. We show that the interplay between doubly dipolar interactions, quantum stabilization, and external confinement results in a rich ground-state physics of supersolids and incoherent droplet arrays in doubly dipolar condensates. Our study reveals novel possibilities for engineering quantum droplets and droplet supersolids, including supersolid-supersolid transitions and the realization of supersolid arrays of pancake droplets.

We investigate the dynamics of a three-dimensional Bose-Einstein condensate of ultracold atomic gases with a soft-core-shaped long-range interaction, which is induced by laser dressing the atoms to a highly excited Rydberg state. For a homogeneous condensate, the long-range interaction drastically alters the dispersion relation of the excitation, supporting both roton and maxon modes. Rotons are typically responsible for the creation of supersolids, while maxons are normally dynamically unstable in BECs with dipolar interactions. We show that maxon modes in the Rydberg-dressed condensate, on the contrary, are dynamically stable. We find that the maxon modes can be excited through an interaction quench, i.e., turning on the soft-core interaction instantaneously. The emergence of the maxon modes is accompanied by oscillations at high frequencies in the quantum depletion, while rotons lead to much slower oscillations. The dynamically stable excitation of the roton and maxon modes leads to persistent oscillations in the quantum depletion. Through a self-consistent Bogoliubov approach, we identify the dependence of the maxon mode on the soft-core interaction. Our study shows that maxon and roton modes can be excited dynamically and simultaneously by quenching Rydberg-dressed long-range interactions. This is relevant to current studies in creating and probing exotic states of matter with ultracold atomic gases.