Special issue on Rydberg atomic physics

We theoretically analyze the interactions and decay rates for atoms dressed by multiple laser fields to strongly interacting Rydberg states using a quantum master equation approach. In this framework a comparison of two-level and three-level Rydberg-dressing schemes is presented. We identify a resonant enhancement of the three-level dressed interaction strength which originates from cooperative multiphoton couplings as well as small distance dependent decay rates. In this regime the soft-core shape of the potential is independent of the sign of the bare Rydberg–Rydberg interaction, while its sign can be repulsive or attractive depending on the intermediate state detuning. As a consequence, near-resonant Rydberg dressing in three-level atomic systems may enable the realization of laser driven quantum fluids with long-range and anisotropic interactions and with controllable dissipation.

In this brief review, the opportunities that the alkaline-earth elements offer for studying new aspects of Rydberg physics are discussed. For example, the bosonic alkaline-earth isotopes have zero nuclear spin which eliminates many of the complexities present in alkali Rydberg atoms, permitting simpler and more direct comparison between theory and experiment. The presence of two valence electrons allows the production of singlet and triplet Rydberg states that can exhibit a variety of attractive or repulsive interactions. The availability of weak intercombination lines is advantageous for laser cooling and for applications such as Rydberg dressing. Excitation of one electron to a Rydberg state leaves behind an optically active core ion allowing, for high-L states, the optical imaging of Rydberg atoms and their (spatial) manipulation using light scattering. The second valence electron offers the possibility of engineering long-lived doubly excited states such as planetary atoms. Recent advances in both theory and experiment are highlighted together with a number of possible directions for the future.

This review summarizes experimental works performed over the last decade by several groups on the manipulation of a few individual interacting Rydberg atoms. These studies establish arrays of single Rydberg atoms as a promising platform for quantum-state engineering, with potential applications to quantum metrology, quantum simulation and quantum information.

By mapping the strong interaction between Rydberg excitations in ultra-cold atomic ensembles onto single photons via electromagnetically induced transparency, it is now possible to realize a medium which exhibits a strong optical nonlinearity at the level of individual photons. We review the theoretical concepts and the experimental state-of-the-art of this exciting new field, and discuss first applications in the field of all-optical quantum information processing.

We discuss the density shift and broadening of Rydberg spectra measured in cold, dense atom clouds in the context of Rydberg atom spectroscopy done at room temperature, dating back to the experiments of Amaldi and Segrè in 1934. We discuss the theory first developed in 1934 by Fermi to model the mean-field density shift and subsequent developments of the theoretical understanding since then. In particular, we present a model whereby the density shift is calculated using a microscopic model in which the configurations of the perturber atoms within the Rydberg orbit are considered. We present spectroscopic measurements of a Rydberg atom, taken in a Bose–Einstein condensate and thermal clouds with densities varying from 5 × 10¹⁴ to 9 × 10¹² cm⁻³. The density shift measured via the spectrum's center of gravity is compared with the mean-field energy shift expected for the effective atom cloud density determined via a time of flight image. Lastly, we present calculations and data demonstrating the ability of localizing the Rydberg excitation via the density shift within a particular density shell for high principal quantum numbers.

We present a review of quantum computation with neutral atom qubits. After an overview of architectural options and approaches to preparing large qubit arrays we examine Rydberg mediated gate protocols and fidelity for two- and multi-qubit interactions. Quantum simulation and Rydberg dressing are alternatives to circuit based quantum computing for exploring many body quantum dynamics. We review the properties of the dressing interaction and provide a quantitative figure of merit for the complexity of the coherent dynamics that can be accessed with dressing. We conclude with a summary of the current status and an outlook for future progress.

We investigate the coherent manipulation of interacting Rydberg atoms placed inside a high-finesse optical cavity for the deterministic preparation of strongly coupled light-matter systems. We consider a four-level diamond scheme with one common Rydberg level for N interacting atoms. One side of the diamond is used to excite the atoms into a collective 'superatom' Rydberg state using either π-pulses or stimulated Raman adiabatic passage (STIRAP) pulses. The upper transition on the other side of the diamond is used to transfer the collective state to one that is coupled to a field mode of an optical cavity. Due to the strong interaction between the atoms in the Rydberg level, the Rydberg blockade mechanism plays a key role in the deterministic quantum state synthesis of the atoms in the cavity. We use numerical simulation to show that non-classical states of light can be generated and that the state that is coupled to the cavity field is a collective one. We also investigate how different decay mechanisms affect this interacting many-body system. We also analyze our system in the case of two Rydberg excitations within the blockade volume. The simulations are carried out with parameters corresponding to realizable high-finesse optical cavities and alkali atoms like rubidium.

The propagation of light fields through a quasi one-dimensional cold atomic gas, exciting atomic Rydberg levels of large principal quantum number under conditions of electromagnetically induced transparency, can lead to a stable two-mode Luttinger liquid system. Atomic van der Waals interactions induce a coupling of bosonic field modes that display both photonic and atomic character, the Rydberg dark-state polaritons (RDPs). It is shown that by tunable control of the van der Waals coupling, the RDP may decouple into independent 'spin' and 'charge' fields which propagate at different speeds, analogous to spin–charge separation of electrons in a one-dimensional metal.

Beams of helium atoms in Rydberg–Stark states with principal quantum number n = 48 and electric dipole moments of 4600 D have been decelerated from a mean initial longitudinal speed of 2000 m s⁻¹ to zero velocity in the laboratory-fixed frame-of-reference in the continuously moving electric traps of a transmission-line decelerator. In this process accelerations up to $-1.3\times {10}^{7}$ m s⁻² were applied, and changes in kinetic energy of ${\rm{\Delta }}{E}_{\mathrm{kin}}=1.3\times {10}^{-20}$ J (${\rm{\Delta }}{E}_{\mathrm{kin}}/e=83$ meV) per atom were achieved. Guided and decelerated atoms, and those confined in stationary electrostatic traps, were detected in situ by pulsed electric field ionisation. The results of numerical calculations of particle trajectories within the decelerator have been used to characterise the observed deceleration efficiencies, and aid in the interpretation of the experimental data.

We consider the excitation of Rydberg states through photons carrying an intrinsic orbital angular momentum degree of freedom. Laguerre–Gauss modes, with a helical wave-front structure, correspond to such a set of laser beams, which carry ${{\ell }}_{0}$ units of orbital angular momentum in their propagation direction, with ℓ₀ the winding number. We demonstrate that, in a proper geometry setting, this orbital angular momentum can be transferred to the internal degrees of freedom of the atoms, thus violating the standard dipole selection rules. Higher orbital angular momentum states become accessible through a single photon excitation process. We investigate how the spacial structure of the Laguerre–Gauss beam affects the radial coupling strength, assuming the simplest case of hydrogen-like wavefunctions. Finally we discuss a generalization of the angular momentum coupling, in order to include the effects of the fine and hyperfine splitting, in the context of the Wigner–Eckart theorem.

We examine the adiabatic preparation of crystalline phases of Rydberg excitations in a one-dimensional lattice gas by frequency sweep of the excitation laser, as proposed by Pohl et al (2010 Phys. Rev. Lett. 104 043002) and recently realized experimentally by Schauß et al (2015 Science 347 1455). We find that the preparation of crystals of a few Rydberg excitations in a unitary system of several tens of atoms requires exceedingly long times for the adiabatic following of the ground state of the system Hamiltonian. Using quantum stochastic (Monte Carlo) wavefunction simulations, we show that realistic decay and dephasing processes affecting the atoms during the preparation lead to a final state of the system that has only a small overlap with the target crystalline state. Yet, the final number and highly sub-Poissonian statistics of Rydberg excitations and their spatial order are little affected by the relaxations.

We experimentally characterize the optical nonlinear response of a cold atomic medium placed inside an optical cavity, and excited to Rydberg states. The excitation to S and D Rydberg levels is carried out via a two-photon transition in an electromagnetically induced transparency configuration, with a weak (red) probe beam on the lower transition, and a strong (blue) coupling beam on the upper transition. The observed optical nonlinearities induced by S states for the probe beam can be explained using a semi-classical model with van der Waals' interactions. For the D states, it appears necessary to take into account a dynamical decay of Rydberg excitations into a long-lived dark state. We show that the measured nonlinearities can be explained by using a Rydberg bubble model with a dynamical decay.

We consider the Grover search algorithm implementation for a quantum register of size $N={2}^{k}$ using k (or $k+1$) microwave- and laser-driven Rydberg-blockaded atoms, following the proposal by Mølmer et al (2011 J. Phys. B 44 184016). We suggest some simplifications for the microwave and laser couplings, and analyze the performance of the algorithm for up to k = 4 multilevel atoms under realistic experimental conditions using quantum stochastic (Monte Carlo) wavefunction simulations.

Trapping a Rydberg atom close to a surface is an important step towards the realisation of many proposals for quantum information processing or hybrid quantum systems. One of the challenges in these experiments is posed by the electric field emanating from contaminations on the surface. Here we report on measurements of an electric field created by ⁸⁷Rb atoms adsorbed on a 25 nm thick layer of SiO₂, covering a 90 nm layer of Au. The electric field is measured using a two-photon transition to the $23{D}_{5/2}$ and $25{S}_{1/2}$ states. The electric field value that we measure is higher than typical values measured above metal surfaces, but is consistent with a recent measurement above a SiO₂ surface. In addition, we measure the temporal behaviour of the field and observe that we can reduce it in a single experimental cycle, using ultraviolet light or by mildly locally heating the surface with one of the excitation lasers, whereas the buildup of the field takes thousands of cycles. We explain these results by a change in the adatom distribution on the surface. These results indicate that, while the stray electric field can be reduced, achieving field-free conditions above a silica-coated gold chip remains challenging.

Supersonic beams of hydrogen atoms, prepared selectively in Rydberg–Stark states of principal quantum number n in the range between 25 and 35, have been deflected by ${90}^{\circ }$, decelerated and loaded into off-axis electric traps at initial densities of $\approx {10}^{6}$ atoms cm⁻³ and translational temperatures of 150 mK. The ability to confine the atoms spatially was exploited to study their decay by radiative and collisional processes. The evolution of the population of trapped atoms was measured for several milliseconds in dependence of the principal quantum number of the initially prepared states, the initial Rydberg-atom density in the trap, and the temperature of the environment of the trap, which could be varied between 7.5 and 300 K using a cryorefrigerator. At room temperature, the population of trapped Rydberg atoms was found to decay faster than expected on the basis of their natural lifetimes, primarily because of absorption and emission stimulated by the thermal radiation field. At the lowest temperatures investigated experimentally, the decay was found to be multiexponential, with an initial rate scaling as ${n}^{-4}$ and corresponding closely to the natural lifetimes of the initially prepared Rydberg–Stark states. The decay rate was found to continually decrease over time and to reach an almost n-independent rate of more than (1 ms)⁻¹ after 3 ms. To analyze the experimentally observed decay of the populations of trapped atoms, numerical simulations were performed which included all radiative processes, i.e., spontaneous emission as well as absorption and emission stimulated by the thermal radiation. These simulations, however, systematically underestimated the population of trapped atoms observed after several milliseconds by almost two orders of magnitude, although they reliably predicted the decay rates of the remaining atoms in the trap. The calculations revealed that the atoms that remain in the trap for the longest times have larger absolute values of the magnetic quantum number m than the optically prepared Rydberg–Stark states, and this observation led to the conclusion that a much more efficient mechanism than a purely radiative one must exist to induce transitions to Rydberg–Stark states of higher $| m|$ values. While searching for such a mechanism, we discovered that resonant dipole–dipole collisions between Rydberg atoms in the trap represent an extremely efficient way of inducing transitions to states of higher $| m|$ values. The efficiency of the mechanism is a consequence of the almost perfectly linear nature of the Stark effect at the moderate field strengths used to trap the atoms, which permits cascades of transitions between entire networks of near-degenerate Rydberg-atom-pair states. To include such cascades of resonant dipole–dipole transitions in the numerical simulations, we have generalized the two-state Förster-type collision model used to describe resonant collisions in ultracold Rydberg gases to a multi-state situation. It is only when considering the combined effects of collisional and radiative processes that the observed decay of the population of Rydberg atoms in the trap could be satisfactorily reproduced for all n values studied experimentally.

A spectroscopic study of Rydberg states of helium (n  = 30 and 45) in magnetic, electric and combined magnetic and electric fields with arbitrary relative orientations of the field vectors is presented. The emphasis is on two special cases where (i) the diamagnetic term is negligible and both paramagnetic Zeeman and Stark effects are linear (n = 30, B ≤ 120 mT and F = 0–78 V cm⁻¹), and (ii) the diamagnetic term is dominant and the Stark effect is linear (n = 45, B = 277 mT and F = 0–8 V cm⁻¹). Both cases correspond to regimes where the interactions induced by the electric and magnetic fields are much weaker than the Coulomb interaction, but much stronger than the spin–orbit interaction. The experimental spectra are compared to spectra calculated by determining the eigenvalues of the Hamiltonian matrix describing helium Rydberg states in the external fields. The spectra and the calculated energy-level diagrams in external fields reveal avoided crossings between levels of different m_l values and pronounced m_l-mixing effects at all angles between the electric- and magnetic-field vectors other than 0. These observations are discussed in the context of the development of a method to generate dense samples of cold atoms and molecules in a magnetic trap following Rydberg–Stark deceleration.

We introduce a method to measure radio frequency (RF) electric fields (E-fields) using atoms contained in a prism-shaped vapor cell. The method utilizes the concept of electromagnetically induced transparency with Rydberg atoms. The RF E-field induces changes in the index of refraction of the vapor resulting in deflection of the probe laser beam as it passes through the prism-shaped vapor cell. We measured a minimum RF E-field of 8.25 $\mu {\mathrm{Vcm}}^{-1}$ with a sensitivity of $\sim 46.5\;\mu {\mathrm{Vcm}}^{-1}\;{\mathrm{Hz}}^{-1/2}$. The experimental results agree with a numerical model that includes dephasing effects. We discuss possible improvements to obtain higher sensitivity for RF E-field measurements.

A wavepacket propagation study is reported for the charge transfer of low principal quantum number (n = 2) hydrogen Rydberg atoms incident at an isolated metallic sphere. Such a sphere acts as a model for a nanoparticle. The three-dimensional confinement of the sphere yields discrete surface-localized 'well-image' states, the energies of which vary with sphere radius. When the Rydberg atom energy is degenerate with one of the quantized nanoparticle states, charge transfer is enhanced, whereas for off-resonant cases little to no charge transfer is observed. Greater variation in charge-transfer probability is seen between the resonant and off-resonant examples in this system than for any other Rydberg-surface system theoretically investigated thus far. The results presented here indicate that it may be possible to use Rydberg-surface ionization as a probe of the surface electronic structure of a nanoparticle, and nanostructures in general.

Sufficiently high densities in Bose–Einstein condensates provide favorable conditions for the production of ultralong-range polyatomic molecules consisting of one Rydberg atom and a number of neutral ground state atoms. The chemical binding properties and electronic wave functions of these exotic molecules are investigated analytically via hybridized diatomic states. The effects of the molecular geometry on the system's properties are studied through comparisons of the adiabatic potential curves and electronic structures for both symmetric and randomly configured molecular geometries. General properties of these molecules with increasing numbers of constituent atoms and in different geometries are presented. These polyatomic states have spectral signatures that lead to non-Lorentzian line-profiles.

In excited molecules, the interaction between the covalent Rydberg and ion-pair channels forms a unique class of excited states, in which the infinite manifold of vibrational levels are the equivalent of atomic Rydberg states with a heavy electron mass. Production of the ion-pair states usually requires excitation through one or several interacting Rydberg states; these interacting channels lead to loss of flux, diminishing the rate of ion-pair production. Here, we develop an analytical, asymptotic charge-transfer model for the interaction between ultracold Rydberg molecular states, and employ this method to demonstrate the utility of off-resonant field control over the ion-pair formation, with near unity efficiency.

We have numerically simulated quantum tomography of single-qubit and two-qubit quantum gates with qubits represented by mesoscopic ensembles containing random numbers of atoms. Such ensembles of strongly interacting atoms in the regime of Rydberg blockade are known as Rydberg superatoms. The stimulated Raman adiabatic passage (STIRAP) in the regime of Rydberg blockade is used for determining Rydberg excitation in the ensemble, required for the storage of quantum information in the collective state of the atomic ensemble and implementation of two-qubit gates. The optimized shapes of the STIRAP pulses are used to achieve high fidelity of the population transfer. Our simulations confirm the validity and high fidelity of single-qubit and two-qubit gates with Rydberg superatoms.

The results of theoretical studies of the resonant quenching and ion-pair formation processes induced by collisions of Rydberg atoms with highly polar molecules possessing small electron affinities are reported. We elaborate an approach for describing collisional dynamics of both processes and demonstrate the predominant role of resonant quenching channel of reaction for the destruction of Rydberg states by electron-attaching molecules. The approach is based on the solution of the coupled differential equations for the transition amplitudes between the ionic and Rydberg covalent terms of a quasimolecule formed during a collision of particles. It takes into account the possibility of the dipole-bound anion decay in the Coulomb field of the positive ionic core and generalizes previous models of charge-transfer processes involving Rydberg atoms to the cases, when the multistate Landau–Zener approaches become inapplicable. Our calculations for ${\rm{Rb}}({nl})$ atom perturbed by ${{\rm{C}}}_{2}{{\rm{H}}}_{4}{\mathrm{SO}}_{3}$, ${\mathrm{CH}}_{2}\mathrm{CHCN}$, ${\mathrm{CH}}_{3}{\mathrm{NO}}_{2}$, ${\mathrm{CH}}_{3}\mathrm{CN}$, ${{\rm{C}}}_{3}{{\rm{H}}}_{2}{{\rm{O}}}_{3}$, and ${{\rm{C}}}_{3}{{\rm{H}}}_{4}{{\rm{O}}}_{3}$ molecules show that the curves representing the dependence of the resonant quenching cross sections on the principal quantum number n are bell-shaped with the positions of maxima being shifted towards lower values of n and the peak values, ${\sigma }_{\mathrm{max}}^{({\rm{q}})}$, several times higher than those for the ion-pair formation, ${\sigma }_{\mathrm{max}}^{({\rm{i}})}$. We obtain a simple power relation between the energy of electron affinity of a molecule and the position of maximum in n-dependence of the resonant quenching cross section. It can be used as an additional means for determining small binding energies of dipole-bound anions from the experimental data on resonant quenching of Rydberg states by highly polar molecules.

We investigate the electronic structure of a triatomic Rydberg molecule formed by a Rydberg atom and two neutral ground-state atoms. Taking into account the s-wave and p-wave interactions, we perform electronic structure calculations and analyze the adiabatic electronic potentials evolving from the Rb $(n=35,l\geqslant 3)$ Rydberg degenerate manifold. We hereby focus on three different classes of geometries of the Rydberg molecules, including symmetric, asymmetric and planar configurations. The metamorphosis of these potential energy surfaces in the presence of an external electric field is explored.

It has been demonstrated that very large optical nonlinearities can arise in cold Rydberg gases from strong Rydberg–Rydberg interactions. The interactions between atoms excited to a degenerate Rydberg level are described by a large number of molecular potentials which greatly complicates the theoretical treatment of these systems. We here present a method for very accurate calculation of the third order interaction-induced optical nonlinearities that fully includes the angle-dependent mixing of molecular states by the control optical field. In addition, we investigate how an effective potential can be introduced to describe the third-order optical susceptibility arising from the underlying multi-potential Rydberg–Rydberg interactions. We show that a single effective potential can replace a manifold of asymptotically degenerate potentials of the same sign. Therefore, one effective potential has to be defined for attractive interactions and another for repulsive ones. As an example, we have calculated effective C₆ coefficients of nd + nd asymptotes of rubidium and cesium. We compare accurately calculated collisional integrals with those obtained using effective potentials.

We propose a setup of an open quantum system in which the environment can be tuned such that either Markovian or non-Markovian system dynamics can be achieved. The implementation uses ultracold Rydberg atoms, relying on their strong long-range interactions. Our suggestion extends the features available for quantum simulators of molecular systems employing Rydberg aggregates and presents a new test bench for fundamental studies of the classification of system–environment interactions and the resulting system dynamics in open quantum systems.

Recent high-resolution absorption spectroscopy on highly excited excitons in cuprous oxide (Kazimierczuk et al 2014 Nature 514 343–347) have revealed significant deviations of their spectrum from the ideal hydrogen-like series. In atomic physics, the influence of the ionic core and the resulting modifications of the Coulomb interaction are accounted for by the introduction of a quantum defect. Here we translate this concept to the realm of semiconductor physics and show how the complex band dispersion of a crystal is mirrored in a set of empirical parameters similar to the quantum defect in atoms. Experimental data collected from high-resolution absorption spectroscopy in electric fields allow us to compare results for multiple angular momentum states of the yellow and even the green exciton series of ${\mathrm{Cu}}_{2}{\rm{O}}$. The agreement between theory and experiment validates our assignment of the quantum defect to the nonparabolicity of the band dispersion.

The giant electro-optical response of Rydberg atoms manifests itself in the emergence of sidebands in the Rydberg excitation spectrum if the atom is exposed to a radio-frequency (RF) electric field. Here we report on the study of RF-dressed Rydberg atoms inside hollow-core photonic crystal fibres, a system that enables the use of low modulation voltages and offers the prospect of miniaturised vapour-based electro-optical devices. Narrow spectroscopic features caused by the RF field are observed for modulation frequencies up to 500 MHz.

Exceptional points are special parameter points in spectra of open quantum systems, at which resonance energies become degenerate and the associated eigenvectors coalesce. Typical examples are Rydberg systems in parallel electric and magnetic fields, for which we solve the Schrödinger equation in a complete basis to calculate the resonances and eigenvectors. Starting from an avoided crossing within the parameter-dependent spectra and using a two-dimensional matrix model, we develop an iterative algorithm to calculate the field strengths and resonance energies of exceptional points and to verify their basic properties. Additionally, we are able to visualise the wave functions of the degenerate states. We report the existence of various exceptional points. For the hydrogen atom these points are in an experimentally inaccessible regime of field strengths. However, excitons in cuprous oxide in parallel electric and magnetic fields, i.e., the corresponding hydrogen analogue in a solid state body, provide a suitable system, where the high-field regime can be reached at much smaller external fields and for which we propose an experiment to detect exceptional points.

We predict the possibility of 'triply magic' optical lattice trapping of neutral divalent atoms. In such a lattice, the ${}^{1}{{\rm{S}}}_{0}$ and ${}^{3}{{\rm{P}}}_{0}$ clock states and an additional Rydberg state experience identical optical potentials, fully mitigating detrimental effects of the motional decoherence. In particular, we show that this triply magic trapping condition can be satisfied for Yb atom at optical wavelengths and for various other divalent systems (Ca, Mg, Hg and Sr) in the UV region. We assess the quality of triple magic trapping conditions by estimating the probability of excitation out of the motional ground state as a result of the excitations between the clock and the Rydberg states. We also calculate trapping laser-induced photoionization rates of divalent Rydberg atoms at magic frequencies. We find that such rates are below the radiative spontaneous-emission rates, due to the presence of Cooper minima in photoionization cross-sections.

We obtain ab initio the Hubbard parameters for Rydberg-dressed atoms in a one-dimensional (1D) sinusoidal optical lattice on the basis of maximally-localized Wannier states. Finite range, soft-core interatomic interactions become the trait of Rydberg admixed atoms, which can be extended over many neighboring lattice sites. In contrast to dipolar gases, where the interactions follow an inverse cubic law, the key feature of Rydberg-dressed interactions is the possibility of making neighboring couplings to the same magnitude as that of the onsite ones. The maximally-localized Wannier functions (MLWFs) are typically calculated via a spread-minimization procedure (Marzari N and Vanderbilt D 1997 Phys. Rev. B 56 12847) and are always found to be real functions apart from a trivial global phase when an isolated set of Bloch bands are considered. For an isolated single Bloch band, the above procedure reduces to a simple quasi-momentum-dependent unitary phase transformation. Here, instead of minimizing the spread, we employ a diagonal phase transformation which eliminates the imaginary part of the Wannier functions. The resulting Wannier states are found to be maximally localized and in exact agreement with those obtained via a spread-minimization procedure. Using these findings, we calculate the Hubbard couplings from the Rydberg admixed interactions, including dominant density-assisted tunneling (DAT) coefficients. Finally, we provide realistic lattice parameters for the state-of-the-art experimental Rydberg-dressed rubidium setup.

This review summarizes experimental works performed over the last decade by several groups on the manipulation of a few individual interacting Rydberg atoms. These studies establish arrays of single Rydberg atoms as a promising platform for quantum-state engineering, with potential applications to quantum metrology, quantum simulation and quantum information.

We report an experimental investigation of the facilitated excitation dynamics in off-resonantly driven Rydberg gases by separating the initial off-resonant excitation phase from the facilitation phase, in which successive facilitation events lead to excitation avalanches. We achieve this by creating a controlled number of initial seed excitations. Greater insight into the avalanche mechanism is obtained from an analysis of the full counting distributions. We also present simple mathematical models and numerical simulations of the excitation avalanches that agree well with our experimental results.

We demonstrate the excitation of ions to the Rydberg state $22F$ by vacuum ultraviolet radiation at a wavelength of 123 nm combined with the coherent manipulation of the optical qubit transition in ${}^{40}{\mathrm{Ca}}^{+}$. With a tightly focused beam at 729 nm wavelength we coherently excite a single ion from a linear string into the metastable $3{D}_{5/2}$ state before a VUV pulse excites it to the Rydberg state. In combination with ion shuttling in the trap, we extend this approach to the addressed excitation of multiple ions. The coherent initialization as well as the addressed Rydberg excitation are key prerequisites for more complex applications of Rydberg ions in quantum simulation or quantum information processing.

Using classical trajectories simulations, we investigate the dynamics of a cold sample of Rydberg atoms with high permanent electric dipole moments. The dipolar state can be created using an adiabatic passage through an avoided crossing between an S-like state and a linear Stark state. The simulations yield the pair-correlation functions (PCF) of the atom samples, which allow us to extract the motion of Rydberg-atom pairs in the many-body system. The results reveal the strength and the anisotropic character of the underlying interaction. The simulation is employed to test the suitability of experimental methods designed to derive interaction parameters from PCF. Insight is obtained about the stability of the method against variation of experimentally relevant parameters. Transient correlations due to interaction-induced heating are observed.

Förster resonances provide a highly flexible tool to tune both the strength and the angular shape of interactions between two Rydberg atoms. We give a detailed explanation about how Förster resonances can be found by searching through a large range of possible quantum number combinations. We apply our search method to SS, SD and DD pair states of ⁸⁷Rb with principal quantum numbers from 30 to 100, taking into account the fine structure splitting of the Rydberg states. We find various strong resonances between atoms with a large difference in principal quantum numbers. We quantify the strength of these resonances by introducing a figure of merit ${\tilde{C}}_{3}$ which is independent of the magnetic quantum numbers and geometry to classify the resonances by interaction strength. We further predict to what extent excitation exchange is possible on different resonances and point out limitations of the coherent hopping process. Finally, we discuss the angular dependence of the dipole–dipole interaction and its tunability near resonances.

We present combined measurements of the spatially resolved optical spectrum and the total excited-atom number in an ultracold gas of three-level atoms under electromagnetically induced transparency conditions involving high-lying Rydberg states. The observed optical transmission of a weak probe laser at the center of the coupling region exhibits a double peaked spectrum as a function of detuning, while the Rydberg atom number shows a comparatively narrow single resonance. By imaging the transmitted light onto a charge-coupled-device camera, we record hundreds of spectra in parallel, which are used to map out the spatial profile of Rabi frequencies of the coupling laser. Using all the information available we can reconstruct the full one-body density matrix of the three-level system, which provides the optical susceptibility and the Rydberg density as a function of spatial position. These results help elucidate the connection between three-level interference phenomena, including the interplay of matter and light degrees of freedom and will facilitate new studies of many-body effects in optically driven Rydberg gases.

The dipole–dipole interaction between two Rydberg atoms depends on the relative orientation of the atoms and on the change in the magnetic quantum number. We simulate the effect of this anisotropy on the energy transport in an amorphous many atom system subject to a homogeneous applied electric field. We consider two experimentally feasible geometries and find that the effects should be measurable in current generation imaging experiments. In both geometries atoms of p character are localized to a small region of space which is immersed in a larger region that is filled with atoms of s character. Energy transfer due to the dipole–dipole interaction can lead to a spread of p character into the region initially occupied by s atoms. Over long timescales the energy transport is confined to the volume near the border of the p region which suggests Anderson localization. We calculate a correlation length of 6.3 μm for one particular geometry.

Since the first observation of electron diffraction in 1927, electrons have been used to probe the structure of matter. High-brightness sources of thermal electrons have recently emerged that are capable of simultaneously providing high spatial resolving power along with ultrafast temporal resolution, however they are yet to demonstrate the holy grail of single-shot diffraction of non-crystalline objects. The development of the cold atom electron source, based around the ionisation of laser cooled atoms, has the potential to contribute to this goal. Electron generation from laser cooled atoms is in its infancy, but in just ten years has moved from a proposal to a source capable of performing single-shot diffraction imaging of crystalline structures. The high brightness, high transverse coherence length, and small energy spread of cold electron sources are also potentially advantageous for applications ranging from seeding of x-ray free-electron lasers and synchrotrons to coherent diffractive imaging and microscopy. In this review we discuss the context which motivates the development of these sources, the operating principles of the source, and recent experimental results. The achievements demonstrated thus far combined with theoretical proposals to alleviate current bottlenecks in development promise a bright future for these sources.

We present a degenerate perturbation analysis in the spin–orbit coupled basis for Rydberg atoms in an optical trap. The perturbation matrix is found to be nearly the same for two states with the same total angular momentum j, and orbital angular momentum number l differing by 1, The same perturbation matrices result in the same state-mixing and energy shift. We also study the dependence of state mixing and energy shift on the periodicity and symmetry of the ponderomotive potentials induced by different optical traps. State mixing in a one-dimensional lattice formed with two counterpropagating Gaussian beams is studied and yields a state-dependent trap depth. We also calculate the state-mixing in an optical trap formed by four parallel, separated and highly focused Gaussian beams.

We study interactions between polaritons, arising when photons strongly couple to collective excitations in an array of two-level atoms trapped in an optical lattice inside a cavity. We consider two types of interactions between atoms: dipolar forces and atomic saturability, which range from hard-core repulsion to Rydberg blockade. We show that, in spite of the underlying repulsion in the subsystem of atomic excitations, saturability induces a broadband bunching of photons for two-polariton scattering states. We interpret this bunching as a result of interference, and trace it back to the mismatch of the quantization volumes for atomic excitations and photons. We also examine bound bipolaritonic states: these include states created by dipolar forces, as well as a gap bipolariton, which forms solely due to saturability effects in the atomic transition. Both types of bound states exhibit strong bunching in the photonic component. We discuss the dependence of bunching on experimentally relevant parameters.

Polarized Autler–Townes (AT) splitting of six-wave mixing (SWM) involving Rydberg atoms is for the first time observed in a thermal vapor cell. By scanning the frequency detuning of the dressing field, AT splitting of Rydberg SWM is compared with that of non-Rydberg SWM with an elliptically polarized probe field. It is demonstrated that the AT spectra are strongly dependent on the interaction between Rydberg atoms. Moreover, AT splitting of SWM is cyclically modulated via a multi-dark state and presented by the corresponding spatial SWM AT splitting images. The theoretical calculations are in good agreement with the experimental results.

Cold Rydberg atoms provide novel approaches to many-body problems and quantum simulation. To introduce the recent work presented in this special issue, we present here a quick history of a half-century research activity in the Rydberg-atom field, focusing our attention on the giant interactions between Rydberg atoms and other atoms. These interactions are the origin of many effects observed with Rydberg atoms: pressure shifts, dipole–dipole energy transfer, and avalanche-ionization. These effects have led to evidence of new bound chemical states, such as trilobites states, many-body effects in frozen Rydberg gases, and the spontaneous formation of ultra-cold plasmas. They open exciting new prospects at the intersection of atomic physics, condensed matter physics, and plasma physics.

In the presence of strong dephasing noise the dynamics of Rydberg gases becomes effectively classical, due to the rapid decay of quantum superpositions between atomic levels. Recently a great deal of attention has been devoted to the stochastic dynamics that emerges in that limit, revealing several interesting features, including kinetically constrained glassy behaviour, self-similarity and aggregation effects. However, the non-equilibrium physics of these systems, in particular in the regime where coherent and dissipative processes contribute on equal footing, is yet far from being understood. To explore this we study the dynamics of a small one-dimensional Rydberg lattice gas subject to dephasing noise by numerically integrating the quantum master equation. We interpolate between the coherent and the strongly dephased regime by defining a generalised concept of a blockade length. We find indications that the main features observed in the strongly dissipative limit persist when the dissipation is not strong enough to annihilate quantum coherences at the dynamically relevant time scales. These features include the existence of a time-dependent Rydberg blockade radius, and a growth of the density of excitations which is compatible with the power-law behaviour expected in the classical limit.

This paper offers a toolbox for characterizing the initial conditions and predicting the evolution of the ultracold plasma that forms after resonant laser preparation of a Rydberg gas entrained in a differentially pumped supersonic molecular beam. The conditions afforded by a skimmed free-jet expansion combined with the geometry of laser excitation, determines the phase-space volume of the excited gas. A hydrodynamic shell model, that accounts for the ellipsoidal spatial distribution of this excitation volume in concert with the deforming effects of dissociative recombination, serves to simulate the ambipolar expansion of this molecular ultracold plasma.

The resonant energy transfer between two close particles, also known as Förster resonance in atomic or biological systems, is usually associated with dipole–dipole interaction. In Rydberg atoms, it is a widely used tool to enhance the interactions between particles. Here, we observe a resonant energy transfer between Rydberg atoms that cannot be attributed to a dipole–dipole interaction, owing to selection rules, and comes instead from an efficient dipole–quadrupole process. We compare the measured probability transfer with a theoretical model including quadrupolar terms and find very good agreement with our measurement. Further studies of those multipolar resonances should probe their dependences on various parameters (quantum numbers, relative orientation of the atoms), and may find some applications in quantum procedures where dipole–dipole resonance cannot be used, for instance where the states of interest have a difference in angular momentum of two.

We demonstrate that through localised Rydberg excitation in a three-dimensional cold atom cloud atomic motion can be rendered directed and nearly confined to a plane, without spatial constraints for the motion of individual atoms. This enables creation and observation of non-adiabatic electronic Rydberg dynamics in atoms accelerated by dipole–dipole interactions under natural conditions. Using the full l = 0, 1 $m=0,\pm 1$ angular momentum state space, our simulations show that conical intersection crossings are clearly evident, both in atomic position information and excited state spectra of the Rydberg system. Hence, flexible Rydberg aggregates suggest themselves for probing quantum chemical effects in experiments on length scales much inflated as compared to a standard molecular situation.