The electron in a Rydberg atom is in a large weakly bound orbit, which makes Rydberg atoms exhibit incredibly exaggerated properties relative to ground state ones.^[2,30] In general, almost all properties of Rydberg atoms show a characteristic scaling with n, or more precisely the effective principal quantum number n^* = n − δ_nlj, where δ_nlj denotes quantum defect^[31] which generally could depend on principal quantum number n, orbital angular momentum quantum number l, and total angular momentum quantum number j. Typical scaling laws for a list of common properties are summarized in Table 1.

For instance, figure 1(a) plots the energy levels of ⁸⁷Rb atom. The distribution of energy levels becomes denser as n increases because the energy difference ΔE between neighboring Rydberg states scales as n⁻³, on the order of GHz when n is around 100. The size of Rydberg atoms scales as ⟨r⟩ ∝ n² a₀ with a₀ the Bohr radius, indicating that the dipole moment of Rydberg atoms also scales as n². Select plots for probability distributions of Rydberg-electron radial wavefunction ψ(r) in the equatorial plane for states with different l are shown in Fig. 1(b). It oscillates with r, and the oscillation frequency decreases as l increases. Especially noteworthy is, when l takes the maximal value l = n − 1, the electron is localized around r = n²a₀, forming a nearly circular orbit and such atoms are called circular Rydberg atoms.

Essentially, all Rydberg atoms are unstable. They decay with a characteristic lifetime τ according to the states they reside in, and eventually can fall to the ground state. Two physical mechanisms dominate the decay process, one is spontaneous emission caused by perturbation of the vacuum electromagnetic fluctuations at 0 K (with a rate τ₀), the other is stimulated emission caused by blackbody radiation (BBR) at finite temperature T (with a rate τ_bb). The total lifetime τ of a Rydberg atom at finite temperature is given by Radiative lifetime The τ₀ of a given Rydberg state n can be calculated by summing up the transition rates of all possible spontaneous emission channels, i.e., where A_nn′ is the Einstein A coefficient for the transition from state n to a lower one n′.^[33] From the perspective of classical electrodynamics, the farther an electron is away from the nucleus, the smaller acceleration it experiences and therefore the smaller radiation rate is. Roughly speaking, the lifetime of a Rydberg atom increases as the principal quantum number n becomes larger. More rigorous analysis shows that the scaling laws between τ₀ and n for different l states are not the same.^[34] For state|nl⟩ with a low orbital quantum number l, there exists a large number of dipole allowed transition channels. The decay rates scale as n⁻⁵ or n^−3[34] for transitions to neighboring Rydberg states or to low-energy states, respectively. The transitions to low-lying states constitute dominant channels of spontaneous emission, so the radiative lifetime is approximately proportional to n³. However, the situation will be different if the atom is in a high-l state, e.g., circular Rydberg state with l = n − 1, whose dominant dipole allowed final state is |n′ = n − 1, l′ = n − 2⟩. The radiative lifetime of such states is proportional to n⁵, which can be much larger than that of low-l states, making circular Rydberg atoms prospective carriers for quantum information.^[35–40]

Blackbody lifetime Under normal circumstances, the effects of room-temperature blackbody radiation on atomic systems can be neglected, because for atoms in the ground and low excited states with large transition frequencies ω, the effective number n_ω of BBR photons at T = 300 K is much smaller than unity.^[34] This effect, however, becomes more dramatic for atoms in Rydberg states because these states have low-frequency transitions with large electric dipole matrix elements and are thus strongly coupled to the BBR.^[33] Different from spontaneous emission transitions, BBRs mainly cause atoms to decay to nearby Rydberg states, because the lower the transition frequencies, the higher the mean number of photons in near resonant modes. When the principal quantum number n is large, we only need to focus on the transitions with ω ≪ k_BT, whose lifetimes are approximately given by τ_bb = 3ħ n²/4α³k_BT (ħ = 1 is assumed in the rest of this paper), with α the fine structure constant. This scaling of τ_bb ∝ n² on n is the same for different l states. Taking ⁸⁷Rb atom at room temperature (T = 300 K) as an example, for Rydberg states with n ∼ 50, the total lifetime is on the order of 50 μs, which suggests MHz-rate operations are required to ensure quantum coherence.

Superradiance The decay mechanisms mentioned above apply to every single excited atom, which is usually a valid scenario for experiments in dilute gases. The total lifetime, however, would decrease in dense gases due to collective and cooperative effects caused by photon exchange interaction between different Rydberg atoms. The enhanced decay rates mostly arise from superradiance,^[41] where photons emitted by excited atoms can cause stimulated emission in other atoms, triggering an avalanche of collective decays. Such superradiant signature has been observed experimentally.^[42]

Since Rydberg atoms possess large dipole moments scaling as n², the resulting dipole–dipole interactions increase quickly as n increases. One of the main advantages of Rydberg interaction lies at its versatile controllability, the flexibilities of controlling its strength, sign, anisotropy, and spatial dependence by choosing appropriate states, supplemented by the ability to turn off the interaction by transferring atoms back to the ground state.^[43] In the following, we will provide a brief introduction of Rydberg–Rydberg interaction and its associated extensions.

In theory, any quantum system, with more than two distinguishable states, can be used to encode a quantum-bit of information. Neutral atoms, similar to trapped ions, from which come a variety of species and quantum states to choose, provide abundant yet identical choices of physical qubits with rich internal quantum states. The different choices of atoms and states are mainly decided based on trade-offs between well isolated states possessing longer coherence time and easily accessible quantum levels for convenient initializations, manipulations, detections, and implementations of quantum gates.

Aided by the developed techniques for laser cooling and optical or magnetic trapping, heavy alkali atoms like Rb and Cs were investigated in earlier experiments.^[20] Hyperfine ground states are used to encode qubits due to their long coherence time and large level spacings up to GHz regime. Moreover, they can be readily initialized and detected with optical pumping and resonance fluorescence. Recently, more atomic species are investigated and new possibilities for higher fidelity qubit operations are demonstrated. Alkali earth atoms like Sr,^[86] with two valence electrons, can be further cooled in an optical trap by narrow-line cooling, leading to further suppressed decoherence caused by atomic motion.^[87] One photon excitation from ground clock states to Rydberg levels usually offers a higher excitation Rabi frequency than two-photon Rydberg excitations of alkali atoms at similar level of laser powers.^[86]

Long coherence time of physical qubits is one of the basic requirements for quantum computation. More than 10⁴ quantum gates within coherence time are needed to meet the threshold of QEC.^[84] Detailed analysis of decay mechanisms of the hyperfine coherence of trapped atoms was carried out by Kuhr et al.^[88] Under nominal circumstances, the atom loss rate is dominated by collisions with background gases and worsened by heating of atoms.^[89] With proper shielding of radio-frequency (RF) noise, the longitudinal relaxation time is constrained by Raman scattering, which can be suppressed by confining atoms at the intensity minimum of blue detuned optical trap.^[90] The transverse relaxation time is affected by a host of dephasing mechanisms: including fluctuation of magnetic field, fluctuation of trap depth and position, while thermal motion of atoms can be suppressed by cooling atoms to motional ground states.^[91,92] Coherence time of hundreds of milliseconds without dynamical decoupling is achieved aided by the technique of magic-intensity trapping,^[93] which nulls out the first order dependence of energy level on trapping light-intensity. A similar technique is used to balance coherence time of mixed isotopes and further prolongs the coherence time close to 1 s.^[94]

When building up a two-qubit quantum gate, Rydberg excitation or dressing is employed, which also demands a long coherence time. Imperfections in coherent optical excitation of Rydberg states were analyzed in detail in Ref. [95]. Major contributions to decoherence come from the finite lifetime due to spontaneous emission and BBR caused transitions. Coherence time up to 27 μs is demonstrated in experiment helped by reduced laser phase noise and implementation of spin-echo protocol.^[96] Decoherence caused by spontaneous emission from intermediate state can be reduced by increasing the laser power with a larger inter-mediate state detuning, or by employing one photon excitation scheme.^[86]

Scalability constitutes one of the central challenges for implementing large-scale quantum computation. A rough estimate shows the number of physical qubits needed for a rudimentary universal quantum computer easily extends to more than 10⁶.^[97] To run a quantum algorithm with an intended circuit depth of 10¹⁰, noisy physical qubits with practical error rate at the level of 10⁻⁴ would most likely seek help from QEC. With thousands of physical qubits typically needed to make up a logical qubit, quantum algorithm of a few hundred of logical qubits could easily blossom into roughly millions of physical qubits required.

From this perspective, neutral atoms maybe the most promising platform for constructing QEC codes, since only weak magnetic dipole–dipole and vdW interactions exist between ground state atoms, making it possible to closely trap many atoms in large scale with a rich variety of configurations as demonstrated in optical tweezer arrays or magnetic trap arrays.^[98]

Neutral atoms are inherently identical, thus the requirement for physical resources like laser frequencies, etc. does not grow much with the scaling up of qubits. However, neutral atom based systems suffer from cross-talks when single-site addressing is not perfectly satisfied during gate operations or spontaneous emissions from imaging atoms are reabsorbed by nearby atoms during qubit measurements.

The most challenging issue for a neutral atom based QC platform is the stochastic nature of atom loading into individual traps. Collisional blockade^[99] limits the loading probability of single atom into a small volume trap site to around 50%. One possible approach to circumvent is to raise the loading probability either by augmenting with blue detuned catalysis light^[100] or using superfluid-Mott insulator transition in optical lattice at the expense of longer experimental cycle time,^[101] or as demonstrated more recently with immersion cooling based on more advanced control and manipulations of trapping potentials.^[102] However, probability for simultaneous loading of a large number of atoms still becomes exponentially small below a critical level for single atom loading efficiency.

A bottom-up approach involves rearranging incompletely filled arrays with moving optical traps (shown in Fig. 5). This approach was independently demonstrated first by several groups in 2016: Kim et al.^[7] prepared 9 atoms in two-dimensional arrays with spatial light modulator; Endres et al.^[8] rearranged 50 atoms in a one-dimensional tweeter array controlled by acoustic-optical deflector (AOD); and Barredo et al.^[9] operated with a comparable number of atoms in a two-dimensional array. In 2018, an arbitrary three-dimensional configuration up to 72 atoms was reported.^[103] Recently, it is shown that by combining moving optical tweezer steered by a 2D AOD with microtraps generated from a microlens array, a large defect-free cluster with more than 100 atoms can be created.^[104] Sorting atoms sequentially can exponentially increase the probability for defect-free loading. Loss of atom can be mitigated by a similar procedure with lost atoms replenished from a reservoir.

For atoms in optical lattices with submicron spaced periodic potentials, it is difficult to individually rearrange atoms with micron-scaled moving optical traps. Nevertheless, rearrangement can be carried out using an alternative scheme based on single-site-resolved state flipping and state-sensitive lattice site translation,^[105,106] which has been demonstrated in one-dimensional polarization-synthesized optical lattice^[107] as well as in three-dimensional optical lattice^[108] with individual addressing capability.

Initialization and state detection in a computation cycle most often involve dissipative processes. Initialization of ground state qubits can be simply performed by optical pumping. An additional step for coherent population transfer is needed for qubit encoding into an arbitrary known state.

State measurement is a crucial element for quantum information read-out as well as measurement-based QC or error correction. One simple way to measure the states of atoms is to resonantly push out all atoms of one state and detect the other state by resonant fluorescence imaging of the remaining atoms. This kind of destructive measurement can achieve a high fidelity of 0.9997,^[109] but cannot discriminate atoms selectively pushed out from the ones lost due to background collision. Moreover, the probability of losing about half the atoms implies one needs to replenish atoms after each measurement, which appreciably increases the cycling time of computation.

To address these issues, in situ nondestructive measurement techniques based on state selective fluorescence are developed for both optical lattice^[110] and optical tweezer^[111] systems. Different from nondestructive measurement often referenced to in quantum computation, which categorizes measurements that project qubit state into the same Hilbert space formed by the qubit basis states without excitation into other atom levels, the “nondestructive” measurement here emphasizes measurement without atom lost, because atoms exited to other levels can be easily reinitialized with optical pumping. Simultaneous low-loss measurements of multiatoms are realized recently.^[112,113] Different from destructive measurements, whose errors mainly come from unwanted atoms not blown out or atoms lost from light scattering or background collision during detection, nondestructive measurement incurs additional error from atom heating during imaging. Much effort has been paid to address trade-offs between more imaging cycles for higher contrast against dark noises from non-ideal photon detectors and atom losses due to consequent heating. Collection lenses with moderately high numerical apertures, deeper optical traps, and careful preparation of polarization states for the probe lights are essential to obtain the optimal results.

In 2016, Boll et al. resolved atoms in two spin states by shifting them into different locations with a high-gradient magnet field, realizing a detection fidelity of 98%^[114] [shown in Fig. 6(a)]. An improved result was achieved by Wu et al. in 2019, who employed a state-dependent optical lattice to separate atoms in two qubit states^[92] [shown in Fig. 6(b)]. This new measurement scheme inherits the benefits of both former approaches: atom cooling with imaging light and non-destructive state detection. It can be easily extended to reconfigurable dipole-trap arrays.

In order to take the full advantage of scalability by atomic qubits, parallel large-scale qubit measurements present another challenge. Single site imaging at the wavelengh scale of optical lattice^[115] is carried out with high numerical aperture lens collecting scattered light onto sensitive detectors such as electron multiplying charge coupled device (EMCCD). During post processing of imaging data, Bayesian inference methods are used instead of threshold state discrimination for better state reconstructions in 2D lattices.^[112]

In spite of the steadily increasing number of atomic qubits and the quantum gate fidelities realized in the state-of-the-art experiments, universal quantum computation with QEC remains a long-term goal, considerably out of reach even with scaled up system based on current platforms. However, on the noisy intermediate-scale quantum (NISQ) level, neutral atom based systems can still accomplish several complex computing tasks and exhibit certain quantum features and advantages.

One of the near-term applications for neutral atom based platforms is to solve complex optimization problems by measuring the final states of time evolved quantum many-body systems. A promising paradigm towards this goal is quantum annealing algorithm (QAA), where the solution to a problem is obtained by adiabatically driving the system towards the instantaneous ground state of the objective Hamiltonian. In 2013, Keating et al. proposed a Rydberg dressing based quantum annealer^[159] which can be used to solve the quadratic unconstrained binary optimization problem. Later on, by utilizing Rydberg-mediated four-body interaction, Glaetzle et al. introduced a universal quantum annealer with all-to-all couplings.^[160] It is then found that anisotropy of the Rydberg interaction can also facilitate the construction of a different universal annealer.^[161] Besides QAA, hybrid quantum-classical variational approaches such as quantum approximate optimization algorithms (QAOA) can also be implemented in Rydberg atom systems. Recently, it is found that 2D neutral atom systems are suitable for solving the maximum independent set problem^[162] and MaxCut problem^[163] based on QAOA.

One of the most important research directions for the Rydberg atom many-body system is to simulate coherent spin models. To this end, atoms are first loaded into a given lattice configuration (or an ensemble) and prepared in a specific initial state as described in Subsections 3.2 and 3.3. Subsequently, the system evolves under coherent driving in the presence of Rydberg interactions, during which the motional degree of freedom for each atom can be regarded as frozen if the evolution time is sufficiently short. In such a frozen gas limit, the dynamics only alters the internal atomic states, which can effectively emulate an interacting spin system.

The state-dependent interaction in Rydberg atom systems makes it a natural platform to implement analog simulation of diverse spin models in condensed matter physics. For example, the Rydberg atom system subjected to vdW interaction can be readily mapped to the quantum Ising model, where each atom constitutes a pseudo-spin 1/2 system consisting of a ground state |g⟩ and a Rydberg state |r⟩. The corresponding Hamiltonian in the rotating frame reads where ${\hat{\sigma }}_{\alpha \beta }^{(i)}=|\alpha {\rangle }_{i}\langle \beta |$ and ${\hat{\sigma }}_{x}^{(i)}={\hat{\sigma }}_{rg}^{(i)}+{\hat{\sigma }}_{gr}^{(i)}$ denote the spin operators for the i-th atom, Ω and Δ respectively denote the Rabi frequency and detuning associated with the coherent laser driving, and V_ij = C₆/|r_i − r_j|⁶ is the vdW interaction between the i-th and the j-th atoms when they are both excited to the Rydberg state. By applying the transformation ${\hat{\sigma }}_{rr}^{(i)}=({\hat{\sigma }}_{z}^{(i)}+1)/2$, one can see that equation (10) is equivalent to the Ising model in the presence of transverse and longitudinal magnetic fields. Such a model Hamiltonian is widely studied,^[165–169] and its simulation accuracy has been carefully examined in a recent experiment.^[170]

To date, a lot of experiments have attempted to realize the above introduced Ising model, and here we briefly review the impressive works carried out in an ordered lattice configuration. In 2012, Schauß et al. realized the above Ising Hamiltonian in a 2D optical lattice and observed Rydberg blockade induced dynamical crystallization at resonant driving (Δ = 0).^[171] Continued studies were reported in Ref. [172], where the system was instead adiabatically driven to its crystalline ground state. The Ising model was also simulated in optical tweezer arrays. In 2016, Labuhnin et al. investigated quench dynamics in a partially filled 2D atom array of arbitrary shape.^[173] A major breakthrough is achieved by combining defect-free preparation of atom array with Rydberg excitation. Based on this technique, Bernien et al. carried out its simulation in a 1D defect-free array with up to 51 atoms.^[174] In this experiment, by adiabatically varying the detuning Δ(t) from positive to negative, they kept the system in the ground state and observed a transition from the disorder phase to a crystalline phase of Z₂ order |rgrg ··· gr⟩, Z₃ order |rggrgg ··· ggr⟩, or Z₄ order |rgggrggg ··· gggr⟩, which was determined by the ratio of blockade radius R_b to lattice constant a [see Fig. 10(a)]. Many other interesting problems are addressed in similar setups, including facilitation and localization dynamics,^[175] establishment of detailed balance during thermalization,^[176] and Kibble–Zurek mechanism in critical dynamics.^[177] Antiferromagnetic correlations are also observed in 2D optical lattice^[178] as well as in 2D reconfigurable tweezer array.^[179] Recently, the 2D tweezer array system is increased to a remarkably large scale ∼ 200 qubits in various lattice geometries.^[180,181]

In addition to direct Rydberg excitation, Rydberg dressing discussed in Subsection 2.3 provides an alternative way to implement the quantum Ising model. In the dressing protocol, two internal ground states are used to encode spin-up and spin-down states and the Ising interaction takes the form of Eq. (9). Coherent many-body dynamics of Ising quantum magnets built up by Rydberg dressing are experimentally studied both in an optical lattice^[64,65] and in an atomic ensemble.^[66]

Resonant dipole–dipole interaction between Rydberg atoms can be used to simulate a new type of many-body spin dynamics. For example, a system containing two dipole coupled Rydberg states |s⟩ and |p⟩ can be mapped to a spin-1/2 XY model where ${\hat{\sigma }}_{+}^{(i)}=|s{\rangle }_{i}\langle p|$ (${\hat{\sigma }}_{-}^{(i)}=|p{\rangle }_{i}\langle s|$) is the raising (lowering) operator for the i-th spin, and J_ij = C₃(θ_ij)/|r_i − r_j|³ is the dipolar interaction with C₃(θ_ij) ∝ 1 − 3cos² θ_ij. In this model, Rydberg interaction manifests itself in a coherent excitation transfer between nearby atoms, which is observed in a three-atom system as reported by Ref. [183]. Such a long-range XY interaction also leads to nontrivial many-body relaxations as observed in an atomic ensemble.^[184] By mapping spin excitations to hard-core bosons, this model can be further used to study symmetry protected topological order of strongly interacting bosons, which has been demonstrated in a staggered atom array configuration shown in Fig. 10(b).^[182] In this work, anisotropy of the interaction facilitates the construction of a Su–Schrieffer–Heeger (SSH) model, which possesses chiral symmetry and can give rise to a single-particle edge state as well as a fourfold ground state degeneracy protected by such a symmetry. Recently, it is shown that dipolar exchange interaction is capable of simulating density dependent Peierls phase,^[185] which establishes an essential step towards observations of many exotic topological phenomena.^[186,187]

Different from analog simulation that starts from the microscopic Hamiltonian as described above, digital quantum simulation is carried out by decomposing total unitary evolution operator $\hat{U}(t)$ to sequences of discrete unitary gates.^[188] Applying digital simulation paradigm to Rydberg atom system makes it feasible to explore a broader range of spin dynamics, such as simulation of multiparticle interactions^{[130,189,190]} and quantum cellular automaton dynamics.^[191,192]

The great success in simulating coherent spin dynamics highlights the potential for Rydberg atom system to implement a practical simulator. In the future, we expect this type of simulation will help us gaining further insights into many-body phenomena of interacting spins, studying spin liquid states^[193] and quantum many-body scars,^[194–197] implementing various spin models via Rydberg dressing,^[68,69,198] observing complex orders in higher-dimension lattices,^[199,200] simulating lattice gauge theory,^[201–204] and high-energy physics,^[205] etc.

In the previous subsection, we focus on spin dynamics in the frozen gas limit. For more realistic situation, by taking the atomic and electronic center of mass motion into consideration, it is possible to investigate associated physics in a broader scope. In this subsection, we introduce several promising strategies along this direction. We focus on many-body physics here, although Rydberg atom system can also help us to gain insights into few-body physics such as ultralong-range molecule state.^[206–209]

First, applying Rydberg dressing into the ensemble of ground-state atoms can enrich properties of ultracold quantum gases. For example, when a Bose–Einstein condensate is off-resonantly dressed to a Rydberg state, its wavefunction Ψ(r) evolves according to a modified Gross–Pitaevskii equation where m is the mass of the atom, g is the contact interaction strength, and U(r − r′) denotes the Rydberg dressing induced nonlocal interaction strength. Inclusion of such a nonlocal interaction term is essential for the creation of a novel roton-maxon excitation spectrum^[60] as well as the formation of a supersolid vortex crystal.^[210] Applying similar dressing scheme into optical lattice can facilitate quantum simulation of the Hubbard model with long-range soft-core interactions [see Fig. 11(a)]. Theoretical studies reveal rich exotic phases of matter emerging from such an extended Hubbard model, such as topological Mott insulators,^[211] cluster Luttinger liquids,^[212] superglass phases,^[213] and supersolidities.^[214,215] For atoms subjected to a synthetic gauge field, Rydberg dressing scheme is capable of simulating fractional quantum Hall phases of bosons.^[216]

When an atom is deeply trapped in a potential, it cannot hop to nearby lattice sites. In this case, distance-dependent Rydberg interaction can modify the vibrational state of an atom in the trap, giving rise to effective phonon dynamics, as shown in Fig. 11(b). Coupling between vibrational and internal degrees of freedom can result in complex excitation dynamics (e.g., dissipative blockade effect studied by Ref. [218]) and provides an approach to generating nonclassical motional states.^[219] Recently, it is found that a state-insensitive phonon swap induced by Rydberg dressed interaction is useful for sympathetic cooling of an atomic quantum simulator.^[220] The interplay between spin and vibrational dynamics can also be used to engineer polaronic quasi-particle excitations^[221] and three-body interactions.^[222] Motional dynamics of a flexible array of Rydberg atoms without external trapping potential is intriguing as well, which for instance can simulate a Newton-cradle-like excitation transport.^[223]

Excitations of atoms to their Rydberg states via ultrashort coherent laser pulses provide an approach to exploring many-body physics with strongly interacting electrons. In such an ultrafast quantum simulator, many Rydberg electrons are excited to orbits far from their cation cores. As a result, wavefunctions of these electrons spatially overlap with each other, which then generates strong many-body correlations induced by Coulomb interactions. Based on this idea, Takei et al. observed coherent many-body electron dynamics in an atomic ensemble by time-domain Ramsey interferometry.^[224] Subsequently, Mizoguchi et al. studied many-electron dynamics with an atomic Mott insulator in an optical lattice.^[217] As observed in their experiment [see Fig. 11(c)], overlap of nearby Rydberg electron wavefunctions leads to drastic change of ion-counting statistics and sharp increase of avalanche ionization, which opens up a way to investigate metal-like phases of Rydberg gases.

In a realistic Rydberg atom system, coherent driving offered by external field often competes with dissipation induced by coupling with environment. Such a controllable driven-dissipative system with strong and nonlocal Rydberg–Rydberg interactions can be used to simulate many-body phenomena distinct from their fully coherent counterparts, e.g., dynamical phase transitions that are far from equilibrium.^[225]

Evolution of such an open many-body system is often governed by the master equation ${\partial }_{t}\hat{\rho }=-{\rm{i}}[\hat{H},\hat{\rho }]+\hat{ {\mathcal L} }\hat{\rho }$, where $\hat{\rho }$ is the density matrix of the system, $\hat{H}$ is the Hamiltonian describing coherent driving and interaction, and $\hat{ {\mathcal L} }$ denotes the Liouvillian operator associated with dissipation. For a simple two-level system with Hamiltonian given by Eq. (10), spontaneous decay of Rydberg states can be described by the Liouvillian term $\hat{ {\mathcal L} }\hat{\rho }=\Gamma \displaystyle {\sum }_{n}[{\hat{\sigma }}_{gr}^{(n)}\hat{\rho }{\hat{\sigma }}_{rg}^{(n)}-\{{\hat{\sigma }}_{rr}^{(n)},\hat{\rho }\}/2]$. Such a depopulation term competes with coherent excitation, which together with Rydberg interaction induced blockade or antiblockade can give rise to highly correlated steady state. Theoretical studies of this model reveal a host of interesting steady-state properties, such as non-equilibrium phase transitions,^[226–229] steady-state crystallization and aggregation,^[230–234] as well as bistabilities and metastabilities.^[235,236] Experimental observations of the associated optical bistability and hysteresis in this system were reported in Refs. [237–240].

Another important type of dissipation is dephasing of Rydberg states induced by thermal motion of atoms or phase noise of the driving laser, whose corresponding Liouvillian term takes the form $\hat{ {\mathcal L} }\hat{\rho }=\gamma \displaystyle {\sum }_{n}[{\hat{\sigma }}_{rr}^{(n)}\hat{\rho }{\hat{\sigma }}_{rr}^{(n)}-\{{\hat{\sigma }}_{rr}^{(n)},\hat{\rho }\}/2]$. Such a pure dephasing usually leads to an uncorrelated steady state, but the relaxation process can be highly nontrivial. For example, in the strong dephasing regime, Rydberg interactions result in kinetically constrained dynamics closely related to soft-matter systems,^[243] where the atomic growth rate for Rydberg excitation is strongly affected by the state of its neighboring atoms. In particular, an already excited atom can facilitate Rydberg excitation of another atom located near the facilitation radius r_fac where the vdW interaction V satisfies Δ + V = 0.^[241,244] Such a condition can result in correlated growth of Rydberg aggregates [see Fig. 12(a)], whose features are observed already.^[245–248] As for systems dominated by Rydberg mediated spin-exchange interactions, dephasing can strongly influence the dynamics of excitation transport, which usually exhibits a crossover behavior from quantum ballistic regime to classical diffusive regime.^{[72,249–252]}

In more realistic cases, both decay and dephasing come into play and a Rydberg excitation can decay to another ground state refrained from being excited again. The cooperation of these processes usually leads to complicated dynamical evolutions and nontrivial steady states, such as epidemic dynamics mimicking the spreading of a disease,^[253] mobile excitation holes in Rydberg superatoms,^[254] and directed percolation (DP) phase transitions.^[255,256] As an important example, DP phase transition behaves as a non-equilibrium phase transition from an absorbing phase with vanishing Rydberg populations to an active phase of finite Rydberg densities and large fluctuations [see Fig. 12(b)]. Experimental signatures of such a phase transition are observed in several setups,^{[242,257,258]} which points to a realistic physical system for testing self-organized criticality.

An efficient single-photon source is indispensable for photonic QIP. While it can be realized by coupling a single emitter (e.g., a quantum dot) with a highly localized optical mode (e.g., a micro-cavity mode), Rydberg atomic ensemble provides an alternative way that works in free space.

The original proposal [see Fig. 13(a)] for Rydberg mediated single-photon generation is based on Rydberg blockade and cooperative spontaneous emission.^[264] As a first step of this proposal, excitation beams Ω₁ and Ω₂ induce an effective driving ${\hat{H}}_{{\rm{d}}}=\displaystyle \int {\rm{d}}{\boldsymbol{r}}\sqrt{\rho ({\boldsymbol{r}})}[{\Omega }_{{\rm{eff}}}({\boldsymbol{r}}){\hat{{\mathcal{S}}}}^{\dagger }({\boldsymbol{r}})+{\rm{H}}.{\rm{c}}.]$ with ρ(r) the atomic density and Ω_eff(r) the effective Rabi frequency. If atoms addressed by the effective coupling Ω_eff(r) are within the blockade radius, the strong interaction $\hat{V}$ [Eq. (13)] only allows for the excitation of a single Rydberg polariton $|R\rangle \propto \displaystyle \int {\rm{d}}{\boldsymbol{r}}\sqrt{\rho ({\boldsymbol{r}})}{\Omega }_{{\rm{eff}}}({\boldsymbol{r}}){\hat{{\mathcal{S}}}}^{\dagger }({\boldsymbol{r}})|G\rangle $, where |G⟩ = |g₁ ··· g_N⟩ denotes the spin-wave vacuum state with all atoms in the ground state |g⟩. Subsequently, a coupling beam Ω₃ transfers the created single Rydberg polariton to an excited state polariton, which can then be efficiently converted into a directed photon in a well-defined mode $\hat{ {\mathcal E} }$ based on cooperative spontaneous emission.^[265]

The above scheme was realized by Dudin and Kuzmich.^[260] In their experiment, a strong antibunching of retrived photons is observed as the principal quantum number n is increased [see Fig. 13(b)], indicative of successful generation of single photons. In addition to Rydberg blockade, they found that interaction induced dephasing can also account for the observed antibunching phenomenon. The high single-photon generation efficiency of this system attracts broad attention. In 2016, Li and Kuzmich combined Rydberg blockade with ground-state storage, significantly increasing the total storage time of the system.^[266] Li et al. employed a similar approach to create two orthogonal spin-wave excitations and observed Hong–Ou–Mandel interference between them.^[267] The single-photon generation schemes are also studied in room-temprature atomic gases^[268] and in steadily improved setups.^[269] The reported single-photon emission efficiency in current experiments remains limited due to the small optical depth within the excitation blockade radius. Recently, Petrosyan and Mφlmer proposed a scheme to map the single-atom excitation to a single photon with dipolar exchange interactions,^[270] which relaxes the need for Rydberg blockade and therefore permits a delocalized excitation that can be retrieved more efficiently.

Besides serving as single-photon source, Rydberg atomic ensemble can also be used to create atom-photon entanglement, which plays a pivotal role in the envisaged quantum internet.^[259] Once single Rydberg polariton is prepared, one can map the state |R⟩|0⟩ to an atom-light entangled state $(|R\rangle |0\rangle +|G\rangle |1\rangle)/\sqrt{2}$ through a “half” retrieval operation, where |0⟩ and |1⟩ denote the Fock states for photons. In 2013, Li, Dudin, and Kuzmich realized such entanglement and indeed observed the violation of Bell-inequality.^[271] The entanglement in the above protocol is encoded in the photonic Fock space, while a polarization entangled scheme usually offers a comparatively better performance in a quantum repeater, as it tolerates a larger propagation phase instability and relaxes the need for photon-number-resolved detectors.^[272] In 2019, Li et al. reported a novel scheme to establish atom–photon entanglement in the polarization basis.^[273] Quite recently, Yang, Liu, and You proposed a scheme to realize atom–photon spin-exchange interaction via Rydberg dressing,^[274] with which the desired atom–photon entanglement can be robustly established without storage or retrieval of photons.

Controlling photons by photons requires a nonlinear medium. The nonlinearities of conventional materials at the level of a few photons are relatively small, so that a large number of photons are usually required to observe nonlinear optical effects. Quantum nonlinear optics open the door to studying the regime where a single photon can significantly affect the state of another photon.^[263] Such a strong nonlinearity is key to all-optical quantum logic operations.

As explained in the beginning of this section, coupling photonic field with Rydberg polariton endows photons with strong nonlinearities. An efficient way to achieve this coupling is to employ electromagnetically induced transparency (EIT) as shown in Fig. 13(c). In a Rydberg EIT system, the quantized photonic field $\hat{ {\mathcal E} }({\boldsymbol{r}})$ and the Rydberg polariton field $\hat{{\mathcal{S}}}({\boldsymbol{r}})$ form a dark state polariton (DSP) $\hat{\Psi }({\boldsymbol{r}})=\cos (\theta)\hat{ {\mathcal E} }({\boldsymbol{r}})-\sin (\theta)\hat{{\mathcal{S}}}({\boldsymbol{r}})$.^[275] As a result, the strong interaction $\hat{V}$ in Eq. (13) for the Rydberg component $\hat{{\mathcal{S}}}({\boldsymbol{r}})$ will induce a nonlocal nonliearity for the photonic field $\hat{ {\mathcal E} }({\boldsymbol{r}})$. The original Rydberg-EIT proposal considers the weakly interacting regime,^[276] where two DSPs passing through each other acquire a global phase. In the strongly interacting regime, the nonlinearity can be attributed to a EIT blockade mechanism:^[277–281] the energy of the Rydberg level |r⟩ is modified by the interaction $\hat{V}$, which can destroy the EIT condition. Specifically, when an atom located at r is excited to the Rydberg state, the level |r⟩ for an atom at r′ will experience a shift V(r − r′), which effectively changes the atom at r′ to a two-level system if V(r − r′) is comparable to or larger than the EIT linewidth γ_EIT = Ω²/|γ − iΔ|. Unlike a EIT process with vanishing susceptibility, transmission of the incoming light can be significantly modified by such an effective two-level system. For γ≫Δ, the nonlinearity takes a dissipative feature: photon transmission within the radius z_b = (C₆γ/Ω²) of Rydberg excitation is strongly suppressed. In the regime of Δ≫γ, photons interact with each other via a dispersive nonlocal interaction over the range z_b = (C₆|Δ|/Ω²).^[278]

In 2010, Pritchard et al. demonstrated dissipative nonlinearity of a Rydberg EIT system.^[261] Figure 13(d) shows their experimental results, where a clear suppression of transmission at the EIT peak appears as the input intensity for the probe field $\hat{ {\mathcal E} }$ is increased. The dispersive nonlinearity is then observed by putting Rydberg atoms in a low-finesse cavity and measuring the interaction induced resonance frequency shift.^[282] Although these pioneering experiments were performed in the classical regime, they demonstrated the underline physics and opened up possibilities to explore quantum nonlinear optics in Rydberg EIT systems.

The quantum regime of Rydberg-EIT nonlinearity was first observed by Peyronel et al. in 2012.^[262] In the experiment, by tightly focusing a probe beam into a waist smaller than the blockade radius z_b, input photons are regulated into a train of single photons, as verified by the suppression of the two-photon correlation function displayed in Fig. 13(e). Dispersive quantum nonlinearity is then realized in a similar setup^[283] and characterized by an attractive interaction between photons and the appearance of an interaction induced phase accumulation.

An important class of quantum logic operation — single-photon switch or transistor, allows for the control of target-photon transmission with a single gate photon.^[284] Figures 14(a) and 14(b) display the switching scheme in a Rydberg EIT system. The gate photon in mode ${\hat{ {\mathcal E} }}_{{\rm{g}}}$ is first stored in an atomic ensemble as a collective excitation in Rydberg state |r″⟩. Target photons in mode ${\hat{ {\mathcal E} }}_{{\rm{t}}}$ involving another Rydberg state |r⟩ are then sent into the atomic ensemble. The transmission of the target photon can be suppressed by the stored gate photon, and a large extinction radio can be obtained when the optical depth d_b within the blockade radius is sufficiently high. This scheme was realized by Baur et al.^[285] in 2014. If the switch can offer a gain, i.e., the number of target photons switched by a single gate photon is above unity, it can work as a single-photon transistor. In 2014, Tiarks et al.^[286] and Gorniaczyk et al.^[287] independently demonstrated this type of single-photon transistor with a gain above 10. The gain of the transistor can be further boosted by tuning near to a Föster resonance.^[288]

In 2016, Murray, Gorshkov, and Pohl^[284] showed that there exist certain fundamental limitations to the above introduced optical switch when retrieval of the gate photon is considered. Furthermore, irreversible photon scattering in this scheme prevents the realization of a quantum switch. To address these issues, new schemes are proposed to realize coherent switch. In 2017, Murray and Pohl^[289] proposed a novel polariton switching mechanism. In this proposal [see Figs. 14(c) and 14(d)], a coherent router-type interaction is established, where the target photon in mode ${\hat{ {\mathcal E} }}_{\to }$ is reflected to mode ${\hat{ {\mathcal E} }}_{\leftarrow }$ by a stored gate photon. In 2019, Yang, Liu, and You^[290] introduced a multi-transverse mode scheme, where coherent control of photonic states is mediated by a coupling blockade mechanism [see Figs. 14(e) and 14(f)]: the gate excitation can switch the target-photon state by blockading the coupling between modes ${\hat{ {\mathcal E} }}_{\circlearrowleft }$ and ${\hat{ {\mathcal E} }}_{\circlearrowright }$.

Deterministic quantum-logic gates for photons can also be realized in Rydberg EIT system. In the dispersive regime Δ≫γ, two photons passing through each other will accumulate a global phase ϕ to the two-particle wavefunction, which can be used to build up a photonic CZ gate by tuning ϕ = π. Such an interaction induced phase was first measured by Firstenberg et al.,^[283] where ϕ ∼ 1 rad was reported. A significant phase ϕ ∼ π was then realized by Tiarks et al. in 2016.^[292] For a complete logic gate, photons need to be encoded in different qubit states, as in the work of Tiarks et al.,^[291] where by encoding quantum information in the polarization basis [see Figs. 15(a) and 15(b)], they demonstrated a photonic C-NOT gate with a fidelity of ${ {\mathcal F} }_{{\rm{CNOT}}}=70 \% $ [see Figs. 15(c) and 15(d)]. There exist several alternative Rydberg-based schemes for realizing a photonic quantum gate, e.g., storing photons in atomic ensembles and utilizing Rydberg interactions between stored spin-wave modes,^[293,294] employing cavities to enhance atom–photon couplings,^[295–298] or using dipolar exchange interaction together with a symmetry-protected mechanism to construct a robust π-phase gate.^[299,300]

Rydberg mediated photonic interaction not only supports the construction of quantum-logic operations, but also facilitates quantum simulation with photons.

In fact, simulating few-body dynamics with interacting photons has attracted wide attention.^{[283,301–309]} Particular interests are paid to photonic molecules — bound states formed by a few photons, which resemble molecule states formed by massive particles subjected to an attractive interaction for bonding. In an atomic ensemble, Rydberg EIT with a large intermediate state detuning Δ endows a photon with an attraction as well as an effective mass, which opens the door for the existence of photonic bound state. In 2013, Firstenberg^[283] observed a two-photon bound state in a Rydberg EIT system [see Fig. 16(a)], backed by strong bunching of the two-photon correlation function shown in Fig. 16(b). Subsequently, Bienias et al. examined scattering resonance in the same system,^[301] and Maghrebi et al. studied a diatomic type photon molecule^[302] formed by a Coulomb-like potential. Three-photon bound state was subsequently studied in Refs. [303,310] which included three-body interactions, and in Ref. [304] which discussed a photonic version of Efimov trimer. The associated experiment was reported by Liang et al.,^[306] in which bunching of the third-order correlation function shown in Fig. 16(b) affirms the existence of three-photon clustering. Recently, Bienias et al. found that Lennard–Jones-like potentials can be created by tuning near to a Förster resonance in a Rydberg EIT configuration,^[308] with which many other exotic few-photon bound states can be observed.

Quantum many-body physics lies at the heart of quantum simulation. Many-body physics of interacting photons are particularly interesting because the delocalized nature of a photon can give rise to phenomenon without counterparts in other platforms [see Fig. 17(a)]. As a prominent example, it is predicted that photons passing through a Rydberg atomic ensemble can form a moving-frame crystal,^[311,312] as shown in Fig. 17(b). Several experiments exploring this multiphoton scattering regime are carried out, including interaction between many photons and a Rydberg superatom,^[313] single-photon subtraction mediated by many-body decoherence,^[314] as well as photon transmission through a dissipative Rydberg atomic ensemble at large input rates.^[315] On the companion side, theoretical treatment of multiphoton scattering in a Rydberg atom ensemble is rather challenging, and several methods are developed to uncover the underlying many-body dynamics. In 2013, Gorshkov, Nath, and Pohl introduced a time-ordering method^[316] to describe scattering of a short pulse in a dissipative Rydberg medium and showed that the output single photon is impure. This method is since generalized to the continuous wave limit with linear EIT loss^[317] and to a situation with arbitrary scattering coefficients.^[274] In 2016, Gullans et al. established an effective field theory to describe many-body dynamics in a dispersive Rydberg medium.^[310] Numerical methods such as matrix product states are also applied to the calculation of scattering dynamics.^[315]