Rydberg blockade sphere persists an intriguing picture by which a number of collective many-body effects caused by the strong Rydberg-Rydberg interactions can be clearly understood and profoundly investigated. In the present work, we develop a new definition for the effective two-atom blockade radius and show that the original spherically shaped blockade surface would be deformed when the real number of atoms increases from two to three. This deformation of blockade sphere reveals spatially anisotropic and shrunken properties which strongly depend on the interatomic distance. In addition, we also study the optimal conditions for the Rydberg antiblockade effect and make predictions for improving the antiblockade efficiency in few-atom systems.
We study the long-range interaction of a single ion with a highly excited ultracold Rydberg atom and report on the direct observation of ion-induced Rydberg excitation blockade mediated over tens of micrometer distances. Our hybrid ion-atom system is directly produced from an ultracold atomic ensemble via near-threshold photo-ionization of a single Rydberg excitation, employing a two-photon scheme which is specifically suited for generating a very low-energy ion. The ion's motion is precisely controlled by small electric fields, which allows us to analyze the blockade mechanism for a range of …
We show that the Rydberg blockade mechanism, which is well known in the case of s states, can be significantly different for p and d states due to the van der Waals couplings between different Rydberg Zeeman sublevels and the presence of a magnetic-field. We show, in particular, the existence of magic distances corresponding to the laser-excitation of a superposition of doubly excited states.
Results of numerical Monte Carlo simulations for the Stark-tuned Förster resonance and dipole blockade between two to five cold rubidium Rydberg atoms in various spatial configurations are presented. The effects of the atoms' spatial uncertainties on the resonance amplitude and spectra are investigated. The feasibility of observing coherent Rabi-like population oscillations at a Förster resonance between two cold Rydberg atoms is analyzed. Spectra and the fidelity of the Rydberg dipole blockade are calculated for various experimental conditions, including nonzero detuning from the Förster reso…
High resolution laser Stark excitation of np (60 < n < 85) Rydberg states of ultra-cold cesium atoms shows an efficient blockade of the excitation attributed to long-range dipole-dipole interaction. The dipole blockade effect is observed as a quenching of the Rydberg excitation depending on the value of the dipole moment induced by the external electric field. Effects of eventual ions which could match the dipole blockade effect are discussed in detail but are ruled out for our experimental conditions. Analytic and Monte-Carlo simulations of the excitation of an ensemble of interacting Rydberg…
We investigate the collective excitation effect in a scheme where three identical Rydberg atoms are arranged in an equilateral triangular lattice. By using a static electric field polarizing the atomic dipoles, the dipole-dipole interactions between two Rydberg atoms are essentially anisotropic and can even disappear in the several special resonance cases. For that fact, we observe collectively enhanced excitation probability of single Rydberg atom in resonant areas in the case of strong blockade, and that of double or triple Rydberg atoms in the case of partial blockade. To give more evidence…
We show that microwave spectroscopy of a dense Rydberg gas trapped on a superconducting atom chip in the dipole blockade regime reveals directly the dipole-dipole many-body interaction energy spectrum. We use this method to investigate the expansion of the Rydberg cloud under the effect of repulsive van der Waals forces and the breakdown of the frozen gas approximation. This study opens a promising route for quantum simulation of many-body systems and quantum information transport in chains of strongly interacting Rydberg atoms.
We propose an experiment to demonstrate a novel blockade mechanism caused by long-range anisotropic interactions in an ultracold dipolar gas composed of the recently observed "butterfly" Rydberg molecules. At the blockade radius, the strong intermolecular interaction between two adjacent molecules shifts their molecular states out of resonance with the photoassociation laser, preventing their simultaneous excitation. When the molecules are prepared in a quasi-one-dimensional (Q1D) trap, the interaction's strength can be tuned via a weak external field. The molecular density thus depends strong…
We present a combined experimental and theoretical study of the effects of Rydberg interactions on Autler-Townes spectra of ultracold gases of atomic strontium. Realizing two-photon Rydberg excitation via a long-lived triplet state allows us to probe the thus far unexplored regime where Rydberg state decay presents the dominant decoherence mechanism. The effects of Rydberg interactions are observed in shifts, asymmetries, and broadening of the measured atom-loss spectra. The experiment is analyzed within a one-body density matrix approach, accounting for interaction-induced level shifts and de…
We study theoretically and experimentally the competing blockade and anti-blockade effects induced by spontaneously generated contaminant Rydberg atoms in driven Rydberg systems. These contaminant atoms provide a source of strong dipole-dipole interactions and play a crucial role in the system's behavior. We study this problem theoretically using two different approaches. The first is a cumulant expansion approximation, in which we ignore third-order and higher connected correlations. Using this approach for the case of resonant drive, a many-body blockade radius picture arises, and we find qu…
We present a detailed analysis of our recent observation of synchronous Rabi oscillations between the electronic ground state and Rydberg states in a mesoscopic ensemble containing roughly 100 ultracold atoms [M. Reetz-Lamour \textit{et al.}, submitted, arXiv:0711.4321]. The mesoscopic cloud is selected out of a sample of laser-cooled Rb atoms by optical pumping. The atoms are coupled to a Rydberg state with principal quantum number around 30 by a two-photon scheme employing flat-top laser beams. The influence of residual spatial intensity fluctuations as well as sources of decoherence such as…
One of the most striking features of the strong interactions between Rydberg atoms is the dipole blockade effect, which allows only a single excitation to the Rydberg state within the volume of the blockade sphere. Here we present a method that spatially visualizes this phenomenon in an inhomogeneous gas of ultra-cold rubidium atoms. In our experiment we scan the position of one of the excitation lasers across the cold cloud and determine the number of Rydberg excitations detected as a function of position. Comparing this distribution to the one obtained for the number of ions created by a two…
Accurate quantum gates are basic elements for building quantum computers. There has been great interest in designing quantum logic gates by using blockade effect of Rydberg atoms recently. The fidelity and operation speed of these gates, however, are fundamentally limited by an intrinsic blockade error. Here we propose a type of quantum gates, which are based on Rydberg blockade effect, yet free from any blockade error. In contrast to the "blocking" method in previous schemes, we use Rydberg energy shift to realize a rational generalized Rabi frequency so that a $π$ phase for one input state o…
We study a novel regime of the Rydberg excitation blockade using highly Stark-shifted, yet long-living, states of Rb atoms subject to electric fields above the classical ionization limit. Such states allow tuning the dipole-dipole interaction strength while their ionization rate can be changed over two orders of magnitude by small variations of the electric field. We demonstrate laser excitation of the interacting Rydberg states followed by their detection using controlled ionization and magnified imaging with high spatial and temporal resolution. Our work reveals the hitherto unexplored possi…
High resolution laser excitation of np Rydberg states of cesium atoms shows a dipole blockade at Förster resonances corresponding to the resonant dipole-dipole energy transfer of the np + np → ns + (n + 1)s reaction. The dipole-dipole interaction can be tuned on and off by the Stark effect, and such a process observed for relatively low n (25 − 41) is promising for quantum gate devices. Both Penning ionization and saturation in the laser excitation can limit the range of observation of the dipole blockade
We perform a comprehensive investigation of the coupling between a Rydberg-dressed atomic gas and an ultra-cold plasma. Using simultaneous time-resolved measurements of both neutral atoms and ions, we show that plasma formation occurs via a Coulomb anti-blockade mechanism, in which background ions DC Stark shift nearby atoms into resonance at specific distances. The result is a highly correlated growth of the Rydberg population that shares some similarities with that previously observed for van der Waals interactions. We show that a rate equation model that couples the laser-driven Rydberg gas…
In the laser excitation of ultracold atoms to Rydberg states, we observe a dramatic suppression caused by van der Waals interactions. This behavior is interpreted as a local excitation blockade: Rydberg atoms strongly inhibit excitation of their neighbors. We measure suppression, relative to isolated atom excitation, by up to a factor of 6.4. The dependence of this suppression on both laser irradiance and atomic density are in good agreement with a mean-field model. These results are an important step towards using ultracold Rydberg atoms in quantum information processing.
We develop a multi-photon adiabatic passage to realize a highly efficient Rydberg excitation in a four-level ladder-type atomic system. The adiabatic passage is based on the existence of a novel quasi-dark state in the cascade excitation system where the frequencies of the lasers are appropriately chirped with time. We also investigate the influence of the interatomic Rydberg interaction on the passage and extend its application to the preparation of anti-blockade Rydberg atom pairs in the Rydberg blockade regime.
We present the experimental demonstration of Rydberg blockade in thermal atomic vapor where the atoms are not necessarily be frozen. We show that not all the interacting atoms but only the atoms with same velocity collectively participate in the blockade process. Using this observation, we formulated a suitable model based on super atom picture to study blockade interaction in thermal vapor. We performed an experiment to measure Rydberg population in rubidium thermal vapor using optical heterodyne detection technique and density dependent suppression of Rydberg population is observed in suitab…
We explore a single-photon approach to Rydberg state excitation and Rydberg blockade. Using detailed theoretical models, we show the feasibility of direct excitation, predict the effect of background electric fields, and calculate the required interatomic distance to observe Rydberg blockade. We then measure and control the electric field environment to enable coherent control of Rydberg states. With this coherent control, we demonstrate Rydberg blockade of two atoms separated by 6.6(3) μm. When compared with the more common two-photon excitation method, this single-photon approach is advantag…
The dipole blockade effect at laser excitation of mesoscopic ensembles of Rydberg atoms lies in the fact that the excitation of one atom to a Rydberg state blocks the excitation of other atoms due to the shift in the collective energy levels of interacting Rydberg atoms. It is used to obtain the entangled qubit states based on single neutral atoms in optical traps. In this paper, we present our experimental results on the observation of the dipole blockade for mesoscopic ensembles of 1-5 atoms when they are detected by the selective field ionization method. We investigated the spectra of the t…
The spin-wave (SW) NOON state is an $N$-particle Fock state with two atomic spin-wave modes maximally entangled. Attributed to the property that the phase is sensitive to collective atomic motion, the SW NOON state can be utilized as a novel atomic interferometer and has promising application in quantum enhanced measurement. In this paper we propose an efficient protocol to deterministically produce the atomic SW NOON state by employing Rydberg blockade. Possible errors in practical manipulations are analyzed. A feasible experimental scheme is suggested. Our scheme is far more efficient than t…
We demonstrate the phenomenon of blockade in two-photon excitations to the Rydberg state in thermal vapor. A technique based on optical heterodyne is used to measure the dispersion of a probe beam far off resonant to the D2 line of rubidium in the presence of a strong laser beam that couples to the Rydberg state via two-photon resonance. Density dependent suppression of the dispersion peak is observed while coupling to the Rydberg state with principal quantum number, n = 60. The experimental observation is explained using the phenomenon of Rydberg blockade. The blockade radius is measured to b…
Using time-resolved measurements, we demonstrate coherent collective Rydberg excitation crossing over into Rydberg blockade in a dense and ultracold gas trapped at a distance of 100 $μ$m from a room-temperature atom chip. We perform Ramsey-type measurements to characterize the coherence. The experimental data are in good agreement with numerical results from a master equation using a mean-field approximation, and with results from a super-atom-based Hamiltonian. These results represent significant progress in exploring a strongly interacting driven Rydberg gas on an atom chip.
The Rydberg blockade mechanism has shown noteworthy promise for scalable quantum computation with neutral atoms. Both qubit states and gate-mediating Rydberg state belong to the same optically-trapped atom. The trapping fields, while being essential, induce detrimental decoherence. Here we theoretically demonstrate that this Stark-induced decoherence may be completely removed using powerful concepts of "magic" optical traps. We analyze "magic" trapping of a prototype three-level system: a Rydberg state along with two qubit states: hyperfine states attached to a J=1/2 ground state. Our numerica…
The laser-excitation of Rydberg atoms in ultracold gases is often described assuming that the atomic motion is frozen during the excitation time. We show that this frozen gas approximation can break down for atoms that are held in optical lattices or microtraps. In particular, we show that the excitation dynamics is in general strongly affected by mechanical forces among the Rydberg atoms as well as the spread of the atomic wavepacket in the confining potential. This causes decoherence in the excitation dynamics - resulting in a dissipative blockade effect - that renders the Rydberg excitation…
The spatial evolution of the Rydberg blockade is studied taking into account Stark-split energy levels across several manifolds. We find that the unexpected restoration of a blockaded Rydberg excitation at small interatomic distances, e.g., experimentally observed by P.Schauss, et al. [Nature 491, 87 (2012)], can be explained by the perturbed energy levels from neighboring manifolds that enter the energy window of excitation defined by the bandwidth of the exciting laser. The same mechanism can also explain why the pair correlation function of Rydberg atoms remains nonzero in the entire region…
We study the spatial performance of the entangling gate proposed by H. Levine et al. (Phys. Rev. Lett. 123, 170503 (2019)). This gate is based on a Rydberg blockade technique and consists of just two global laser pulses which drive nearby atoms. We analyze the multilevel Zeeman structure of interacting $^{87}$Rb Rydberg atoms and model two experimentally available excitation schemes using specific driving beams geometry and polarization. In particular, we estimate the blockade shift dependence on inter-atomic distance and angle with respect to the quantization axis. In addition, we show that u…
Two ultracold atom clouds, each separately in a dipole-blockade regime, realize a source of entangled atom pairs that can be ejected on demand. Entanglement generation and ejection is due to resonant dipole-dipole interactions, while van-der-Waals interactions are predominantly responsible for the blockade that ensures the ejection of a single atom per cloud. A source of entangled atoms using these effects can operate with a 10 kHz repetition rate producing ejected atoms with velocities of about 0.5 m/s.
We study the effect of resonances associated with complex molecular interaction of Rydberg atoms on Rydberg blockade. We show that densely-spaced molecular potentials between doubly-excited atomic pairs become unavoidably resonant with the optical excitation at short interatomic separations. Such molecular resonances limit the coherent control of individual excitations in Rydberg blockade. As an illustration, we compute the molecular interaction potentials of Rb atoms near the $100s$ states asymptote to characterize such detrimental molecular resonances, determine the resonant loss rate to mol…
We show that the use of shaped pulses improves the fidelity of a Rydberg blockade two-qubit entangling gate by several orders of magnitude compared to previous protocols based on square pulses or optimal control pulses. Using analytical Derivative Removal by Adiabatic Gate (DRAG) pulses that reduce excitation of primary leakage states and an analytical method of finding the optimal Rydberg blockade we generate Bell states with a fidelity of $F>0.9999$ in a 300 K environment for a gate time of only $50\;{\rm ns}$, which is an order of magnitude faster than previous protocols. These results esta…
By coupling a probe transition to a Rydberg state using electro-magnetically induced transparency (EIT) we map the strong dipole-dipole interactions onto an optical field. We characterize the resulting cooperative optical non-linearity as a function of probe strength and density. We show that the effect of dipole blockade cannot be described using a mean-field but requires an $N$-atom cooperative model. Good quantitative agreement is obtained for three atoms per blockade with the $n=60$ Rydberg state. We place an upper-limit on the dephasing rate of the blockade spheres of $<110$ kHz.
We demonstrate the interaction-induced blockade effect in an ultracold $^{88}$Sr gas via studying the time dynamics of a two-photon excitation to the triplet Rydberg series $5\mathrm{s}n\mathrm{s}\, ^3\textrm{S}_1$ for five different principle quantum numbers $n$ ranging from 19 to 37. By using a multi-pulse excitation sequence to increase the detection sensitivity we could identify Rydberg-excitation-induced atom losses as low as $<1\%$. Based on an optical Bloch equation formalism, treating the Rydberg-Rydberg interaction on a mean-field level, the van der Waals coefficients are extracted fr…
We review here the studies performed about interactions in an assembly of cold Rydberg atoms. We focus more specially the review on the dipole-dipole interactions and on the effect of the dipole blockade in the laser Rydberg excitation, which offers attractive possibilities for quantum engineering. We present first the various interactions between Rydberg atoms. The laser Rydberg excitation of such an assembly is then described with the introduction of the dipole blockade phenomenon. We report recent experiments performed in this subject by starting with the case of a pair of atoms allowing th…
We show that Jaynes-Cummings dynamics can be observed in mesoscopic atomic ensembles interacting with a classical electromagnetic field in the regime of Rydberg blockade, where the time dynamics of the average number of Rydberg excitations in mesoscopic ensembles displays collapses and revivals typical of this model. As the frequency of Rabi oscillations between collective states of Rydberg blockaded ensembles depends on the number of interacting atoms, for randomly loaded optical dipole traps we predict collapses and revivals of Rabi oscillations. We have studied the effects of finite interac…
We propose an efficient method to filter out single atoms from trapped ensembles with unknown number of atoms. The method employs stimulated adiabatic passage to reversibly transfer a single atom to the Rydberg state which blocks subsequent Rydberg excitation of all the other atoms within the ensemble. This triggers the excitation of Rydberg blockaded atoms to short lived intermediate states and their subsequent decay to untrapped states. Using an auxiliary microwave field to carefully engineer the dissipation, we obtain a nearly deterministic single-atom source. Our method is applicable to sm…
We propose to use adiabatic rapid passage with a chirped laser pulse in the strong dipole blockade regime to deterministically excite only one Rydberg atom from randomly loaded optical dipole traps or optical lattices. The chirped laser excitation is shown to be insensitive to the random number \textit{N} of the atoms in the traps. Our method overcomes the problem of the $\sqrt {N} $ dependence of the collective Rabi frequency, which was the main obstacle for deterministic single-atom excitation in the ensembles with unknown \textit{N}, and can be applied for single-atom loading of dipole trap…
Controlling strongly interacting many-body systems enables the creation of tailored quantum matter, with properties transcending those based solely on single particle physics. Atomic ensembles which are optically driven to a Rydberg state provide many examples of this, such as atom-atom entanglement, many-body Rabi oscillations, strong photon-photon interaction and spatial pair correlations. In its most basic form, Rydberg quantum matter consists of an isolated ensemble of strongly interacting atoms spatially confined to the blockade volume - a so-called superatom. Here we demonstrate the cont…
Long range Rydberg blockade interactions have the potential for efficient implementation of quantum gates between multiple atoms. Here we present and analyze a protocol for implementation of a $k$-atom controlled NOT (C$_k$NOT) neutral atom gate. This gate can be implemented using sequential or simultaneous addressing of the control atoms which requires only $2k+3$ or 5 Rydberg $π$ pulses respectively. A detailed error analysis relevant for implementations based on alkali atom Rydberg states is provided which shows that gate errors less than 10% are possible for $k=35$.
We use coherent excitation of 3-16 atom ensembles to demonstrate collective Rabi flopping mediated by Rydberg blockade. Using calibrated atom number measurements, we quantitatively confirm the expected $\sqrt{N}$ Rabi frequency enhancement to within 4%. The resulting atom number distributions are consistent with essentially perfect blockade. We then use collective Rabi $π$ pulses to produce ${\cal N}=1,2$ atom number Fock states with fidelities of 62% and 48% respectively. The ${\cal N}=2$ Fock state shows the collective Rabi frequency enhancement without corruption from atom number fluctuatio…
We study the Rydberg blockade in a system of three atoms arranged in different 2D geometries (linear and triangular configurations). In the strong blockade regime, we observe high-contrast, coherent collective oscillations of the single excitation probability, and an almost perfect van der Waals blockade. Our data is consistent with a total population in doubly and triply excited states below 2%. In the partial blockade regime, we directly observe the anisotropy of the van der Waals interactions between $|nD\rangle$ Rydberg states in the triangular configuration. A simple model, that only uses…
The non-linear and non-local effects in atomic Rydberg media under electromagnetically induced transparency (EIT) make it a versatile platform for fundamental studies and applications in quantum information. In this paper, we study the dynamics of a Rydberg-EIT system in an ensemble that allows for more than one Rydberg excitation in the propagation direction. The density of two-level atoms is such that transient superradiant effects occur. We experimentally observe a cross-over between coherent collective emission (`flash') of two-level atoms to a Rydberg dressed regime ('dressed flash') unde…
We demonstrate the first deterministic entanglement of two individually addressed neutral atoms using a Rydberg blockade mediated controlled-NOT gate. Parity oscillation measurements reveal an entanglement fidelity of $F=0.58\pm0.04$, which is above the entanglement threshold of $F=0.5$, without any correction for atom loss, and $F=0.71\pm0.05$ after correcting for background collisional losses. The fidelity results are shown to be in good agreement with a detailed error model.
Trapped Rydberg ions represent a flexible platform for quantum simulation and information processing which combines a high degree of control over electronic and vibrational degrees of freedom. The possibility to individually excite ions to high-lying Rydberg levels provides a system where strong and long-range interactions between pairs of excited ions can be engineered and tuned via external laser fields. We show that the coupling between Rydberg pair interactions and collective motional modes gives rise to effective long-range multi-body interactions, consisting of two, three, and four-body …
Interaction between Rydberg atoms can significantly modify Rydberg excitation dynamics. Under a resonant driving field the Rydberg-Rydberg interaction in high-lying states can induce shifts in the atomic resonance such that a secondary Rydberg excitation becomes unlikely leading to the Rydberg blockade effect. In a related effect, off-resonant coupling of light to Rydberg states of atoms contributes to the Rydberg anti-blockade effect where the Rydberg interaction creates a resonant condition that promotes a secondary excitation in a Rydberg atomic gas. Here, we study the light-matter interact…
We propose and analyze a new scheme to produce ultracold neutral plasmas deep in the strongly coupled regime. The method exploits the interaction blockade between cold atoms excited to high-lying Rydberg states and therefore does not require substantial extensions of current ultracold plasma experiments. Extensive simulations reveal a universal behavior of the resulting Coulomb coupling parameter, providing a direct connection between the physics of strongly correlated Rydberg gases and ultracold plasmas. The approach is shown to reduce currently accessible temperatures by more than an order o…
We demonstrate $|W\rangle$ state encoding of multi-atom ensemble qubits. Using optically trapped Rb atoms the $T_2$ coherence time is 2.6(3) ms for $\bar N=7.6$ atoms and scales approximately inversely with the number of atoms. Strong Rydberg blockade between two ensemble qubits is demonstrated with a fidelity of $0.89(1)$ and a fidelity of $\sim \hspace{-.05cm}1.0$ when postselected on control ensemble excitation. These results are a significant step towards deterministic entanglement of atomic ensembles.
We calculate interspecies Rydberg-Rydberg interaction strengths for the heavy alkalis Rb and Cs. The presence of strong Förster resonances makes interspecies coupling a promising approach for long range entanglement generation. We also provide an overview of the strongest Förster resonances for Rb-Rb and Cs-Cs using different principal quantum numbers for the two atoms. We show how interspecies coupling can be used for high fidelity quantum non demolition state measurements with low crosstalk in qubit arrays.
The Rydberg excitation blockade has been at the heart of an impressive array of recent achievements; however, state-mixing interactions can compromise its efficiency. When ultracold atoms are excited to Rydberg states near Förster resonance, up to $\sim 50\%$ of atoms can be found in dipole coupled product states within tens of ns after excitation. There has been disagreement in the literature regarding the mechanism by which this mixing occurs. We use state-selective field ionization spectroscopy to measure, on a shot-by-shot basis, the distribution of Rydberg states populated during narrowba…
We present experimental results on two-qubit Rydberg blockade quantum gates and entanglement in a two-dimensional qubit array. Without post selection against atom loss we achieve a Bell state fidelity of $0.73\pm 0.05$, the highest value reported to date. The experiments are performed in an array of single Cs atom qubits with a site to site spacing of $3.8 ~ μ\rm m$. Using the standard protocol for a Rydberg blockade C$_Z$ gate together with single qubit operations we create Bell states and measure their fidelity using parity oscillations. We analyze the role of AC Stark shifts that occur when…
A spatial structure of the zone blocked by the dipolar electric field of a Rydberg atom is calculated taking into account a possibility of excitation to the states with neighboring values of the principal quantum number. As a result, it was found that the blocked zone represents a number of co-centric spherical shells rather than a solid ball, and the respective pair correlation function should have additional maxima at small interparticle distances.
Recently, the notion of two-qubit controlled phase gate via off-resonant modulated driving has been introduced into the neutral atom qubit platform, with respect to both single-photon and two-photon ground-Rydberg transitions. In order to reach a better performance practically, further developments are in need to overcome a few known limitations in previous discussions of this promising method. Here, we thoroughly analyze a variety of modulation styles for two-photon transitions, demonstrating the versatility of off-resonant modulated driving protocols. Furthermore, we show that it is possible…
We present a new mechanism contributing to the detection of photoassociated long-range Rydberg molecules via pulsed-field ionization: ionic products, created by the decay of a long-range Rydberg molecule, modify the excitation spectrum of surrounding ground-state atoms and facilitate the excitation of further atoms into Rydberg states by the photoassociation light. Such an ion-mediated excitation mechanism has been previously called "Coulomb anti-blockade". Pulsed-field ionisation typically doesn't discriminate between the ionization of a long-range Rydberg molecule and an isolated Rydberg ato…
