ARydberg atom is anexcited atom with one or moreelectrons that have a very highprincipal quantum number,n.^[1][2] The higher the value ofn, the farther the electron is from the nucleus,on average. Rydbergatoms have a number of peculiar properties including an exaggerated response toelectric andmagnetic fields,^[3] long decay periods andelectronwavefunctions that approximate, under some conditions,classical orbits of electrons about thenuclei.^[4] The core electrons shield theouter electron from the electric field of the nucleus such that, from a distance, theelectric potential looks identical to that experienced by the electron in ahydrogen atom.^[5]

In spite of its shortcomings, theBohr model of the atom is useful in explaining these properties. Classically, an electron in a circular orbit of radiusr, about a hydrogennucleus of charge +e, obeysNewton's second law:

${\displaystyle \mathbf{F}=m\mathbf{a}\Rightarrow \frac{k{e}^2}{r^2}=\frac{m{v}^2}{r}}$

wherek = 1/(4πε₀).

Orbital momentum isquantized in units ofħ:

${\displaystyle mvr=n\hslash }$.

Combining these two equations leads toBohr's expression for the orbital radius in terms of theprincipal quantum number,n:

${\displaystyle r=\frac{n^2{\hslash }^2}{k{e}^{2}m}.}$

It is now apparent why Rydberg atoms have such peculiar properties: the radius of the orbit scales asn² (then = 137 state of hydrogen has an atomic radius ~1 μm) and the geometric cross-section asn⁴. Thus, Rydberg atoms are extremely large, with loosely boundvalence electrons, easily perturbed orionized by collisions or external fields.

Because thebinding energy of a Rydberg electron is proportional to 1/r and hence falls off like 1/n², the energy level spacing falls off like 1/n³ leading to ever more closely spaced levels converging on the firstionization energy. These closely spaced Rydberg states form what is commonly referred to as theRydberg series.Figure 2 shows some of the energy levels of the lowest three values oforbital angular momentum inlithium.

The existence of the Rydberg series was first demonstrated in 1885 whenJohann Balmer discovered asimple empirical formula for thewavelengths of light associated with transitions in atomichydrogen. Three years later, the Swedish physicistJohannes Rydberg presented a generalized and more intuitive version of Balmer's formula that came to be known as theRydberg formula. This formula indicated the existence of an infinite series of ever more closely spaced discreteenergy levels converging on a finite limit.^[6]

This series was qualitatively explained in 1913 byNiels Bohr with hissemiclassical model of the hydrogen atom in whichquantized values of angular momentum lead to the observed discrete energy levels.^[7][8] A full quantitative derivation of the observed spectrum was derived byWolfgang Pauli in 1926 following development ofquantum mechanics byWerner Heisenberg and others.

The only truly stable state of ahydrogen-like atom is the ground state withn = 1. The study of Rydberg states requires a reliable technique for exciting ground state atoms to states with a large value ofn.

Much early experimental work on Rydberg atoms relied on the use of collimated beams of fast electrons incident on ground-state atoms.^[9]Inelastic scattering processes can use the electronkinetic energy to increase the atoms' internal energy exciting to a broad range of different states including many high-lying Rydberg states,

${\displaystyle e^{-}+A\to A^{\ast }+e^{-}.}$

Because the electron can retain any arbitrary amount of its initial kinetic energy, this process results in a population with a broad spread of different energies.

Another mainstay of early Rydberg atom experiments relied on charge exchange between a beam ofions and a population of neutral atoms of another species, resulting in the formation of a beam of highly excited atoms,^[10]

${\displaystyle A^{+}+B\to A^{\ast }+B^{+}.}$

Again, because the kinetic energy of the interaction can contribute to the final internal energies of the constituents, this technique populates a broad range of energy levels.

The arrival of tunabledye lasers in the 1970s allowed a much greater level of control over populations of excited atoms. In optical excitation, the incidentphoton is absorbed by the target atom, resulting in a precise final state energy. The problem of producing single state, mono-energetic populations of Rydberg atoms thus becomes the somewhat simpler problem of precisely controlling the frequency of the laser output,

${\displaystyle A+\gamma \to A^{\ast }.}$

This form of direct optical excitation is generally limited to experiments with thealkali metals, because the ground statebinding energy in other species is generally too high to be accessible with most laser systems.

For atoms with a largevalence electronbinding energy (equivalent to a large firstionization energy), the excited states of the Rydberg series are inaccessible with conventional laser systems. Initial collisional excitation can make up the energy shortfall allowing optical excitation to be used to select the final state. Although the initial step excites to a broad range of intermediate states, the precision inherent in the optical excitation process means that the laser light only interacts with a specific subset of atoms in a particular state, exciting to the chosen final state.

An atom in aRydberg state has avalence electron in a large orbit far from the ion core; in such an orbit, the outermost electron feels an almosthydrogenic Coulombpotential,U_C, from a compact ion core consisting of anucleus withZprotons and the lower electron shells filled withZ-1 electrons. An electron in the spherically symmetric Coulomb potential has potential energy:

${\displaystyle U_{\text{C}}=-{\displaystyle \frac{e^2}{4\pi {\epsilon }_{0}r}}.}$

The similarity of the effective potential "seen" by the outer electron to the hydrogen potential is a defining characteristic ofRydberg states and explains why the electron wavefunctions approximate to classical orbits in the limit of thecorrespondence principle.^[11] In other words, the electron's orbit resembles the orbit of planets inside a solar system, similar to what was seen in the obsolete but visually usefulBohr andRutherford models of the atom.

There are three notable exceptions that can be characterized by the additional term added to the potential energy:

• An atom may have two (or more) electrons in highly excited states with comparable orbital radii. In this case, the electron-electron interaction gives rise to a significant deviation from the hydrogen potential.^[12] For an atom in a multiple Rydberg state, the additional term,U_ee, includes a summation of eachpair of highly excited electrons:

• If the valence electron has very low angular momentum (interpreted classically as an extremelyeccentric elliptical orbit), then it may pass close enough to polarise the ion core, giving rise to a 1/r⁴ core polarization term in the potential.^[13] The interaction between aninduceddipole and the charge that produces it is always attractive so this contribution is always negative,

${\displaystyle U_{\text{pol}}=-\frac{e^2{\alpha }_{\text{d}}}{(4\pi {\epsilon }_0{)}^2{r}^4},}$

whereα_d is the dipolepolarizability.Figure 3 shows how the polarization term modifies the potential close to the nucleus.

• If the outer electron penetrates the inner electron shells, it will "see" more of the charge of the nucleus and hence experience a greater force. In general, the modification to the potential energy is not simple to calculate and must be based on knowledge of the geometry of the ion core.^[14]

Quantum-mechanically, a state with abnormally highn refers to an atom in which the valence electron(s) have been excited into a formerly unpopulatedelectron orbital with higher energy and lowerbinding energy. In hydrogen the binding energy is given by:

${\displaystyle E_{\text{B}}=-\frac{\mathrm{R}\mathrm{y}}{n^2},}$

where Ry = 13.6eV is theRydberg constant. The low binding energy at high values ofn explains why Rydberg states are susceptible to ionization.

Additional terms in the potential energy expression for a Rydberg state, on top of the hydrogenic Coulomb potential energy require the introduction of aquantum defect,^[5]δ_ℓ, into the expression for the binding energy:

${\displaystyle E_{\text{B}}=-\frac{\mathrm{R}\mathrm{y}}{(n-{\delta }_l{)}^2}.}$

The long lifetimes of Rydberg states with high orbital angular momentum can be explained in terms of the overlapping of wavefunctions. The wavefunction of an electron in a highℓ state (high angular momentum, "circular orbit") has very little overlap with the wavefunctions of the inner electrons and hence remains relatively unperturbed.

The three exceptions to the definition of a Rydberg atom as an atom with a hydrogenic potential, have an alternative, quantum mechanical description that can be characterized by the additional term(s) in the atomicHamiltonian:

• If a second electron is excited into a staten_i, energetically close to the state of the outer electronn_o, then its wavefunction becomes almost as large as the first (a double Rydberg state). This occurs asn_i approachesn_o and leads to a condition where the size of the two electron's orbits are related;^[12] a condition sometimes referred to asradial correlation.^[1] An electron-electron repulsion term must be included in the atomic Hamiltonian.
• Polarization of the ion core produces ananisotropic potential that causes anangular correlation between the motions of the two outermost electrons.^[1][15] This can be thought of as atidal locking effect due to a non-spherically symmetric potential. A core polarization term must be included in the atomic Hamiltonian.
• The wavefunction of the outer electron in states with low orbital angular momentumℓ, is periodically localised within the shells of inner electrons and interacts with the full charge of the nucleus.^[14]Figure 4 shows asemi-classical interpretation of angular momentum states in an electron orbital, illustrating that low-ℓ states pass closer to the nucleus potentially penetrating the ion core. A core penetration term must be added to the atomic Hamiltonian.

Figure 5. Computed energy level spectra of hydrogen in an electric field nearn=15.^[16] The potential energy found in the electronic Hamiltonian for hydrogen is the 1/r Coulomb potential (there is no quantum defect) which does not couple the different Stark states. Consequently the energy levels from adjacentn-manifolds cross at the Inglis–Teller limit.

Figure 6. Computed energy level spectra of lithium in an electric field nearn=15.^[16] The presence of an ion-core that can be polarized and penetrated by the Rydberg electron adds additional terms to the electronic Hamiltonian (resulting in a finite quantum defect) leading to coupling of the different Stark states and henceavoided crossings of the energy levels.

The large separation between the electron and ion-core in a Rydberg atom makes possible an extremely largeelectric dipole moment,d. There is an energy associated with the presence of an electric dipole in anelectric field,F, known in atomic physics as aStark shift,

${\displaystyle E_{\text{S}}=-\mathbf{d}\cdot \mathbf{F}.}$

Depending on the sign of the projection of the dipole moment onto the local electric field vector, a state may have energy that increases or decreases with field strength (low-field and high-field seeking states respectively). The narrow spacing between adjacentn-levels in the Rydberg series means that states can approachdegeneracy even for relatively modest field strengths. The theoretical field strength at which a crossing would occur assuming no coupling between the states is given by theInglis–Teller limit,^[17]

${\displaystyle F_{\text{IT}}={\displaystyle \frac{e}{12\pi {\epsilon }_0{a}_0^2{n}^5}}.}$

In thehydrogen atom, the pure 1/r Coulomb potential does not couple Stark states from adjacentn-manifolds resulting in real crossings as shown infigure 5. The presence of additional terms in the potential energy can lead to coupling resulting in avoided crossings as shown forlithium infigure 6.

The radiative decay lifetimes of atoms in metastable states to the ground state are important to understanding astrophysics observations and tests of the standard model.^[18]

The large sizes and low binding energies of Rydberg atoms lead to a highmagnetic susceptibility,${\displaystyle \chi }$. As diamagnetic effects scale with the area of the orbit and the area is proportional to the radius squared (A ∝n⁴), effects impossible to detect in ground state atoms become obvious in Rydberg atoms, which demonstrate very large diamagnetic shifts.^[19]

Rydberg atoms exhibit strong electric-dipole coupling of the atoms to electromagnetic fields and has been used to detect radio communications.^[20][21]

Rydberg atoms form commonly inplasmas due to the recombination of electrons and positive ions; low energy recombination results in fairly stable Rydberg atoms, while recombination of electrons and positive ions with highkinetic energy often formautoionising Rydberg states. Rydberg atoms' large sizes and susceptibility to perturbation and ionisation by electric and magnetic fields, are an important factor determining the properties of plasmas.^[22]

Condensation of Rydberg atoms formsRydberg matter, most often observed in form of long-lived clusters. The de-excitation is significantly impeded in Rydberg matter by exchange-correlation effects in the non-uniform electron liquid formed on condensation by the collective valence electrons, which causes extended lifetime of clusters.^[23]

Rydberg atoms occur in space due to the dynamic equilibrium betweenphotoionization by hot stars andrecombination with electrons, which at these very low densities usually proceeds via the electron re-joining the atom in a very highn state, and then gradually dropping through the energy levels to the ground state, giving rise to a sequence of recombinationspectral lines spread across theelectromagnetic spectrum. The very small differences in energy between Rydberg states differing inn by one or a few means thatphotons emitted in transitions between such states have low frequencies and long wavelengths, even up toradio waves. The first detection of such a radio recombination line (RRL) was bySovietradio astronomers in 1964; the line, designated H90α, was emitted by hydrogen atoms in then = 90 state.^[24] Today, Rydberg atoms of hydrogen, helium and carbon in space are routinely observed via RRLs, the brightest of which are the Hnα lines corresponding to transitions fromn+1 ton. Weaker lines, Hnβ and Hnγ, withΔn = 2 and 3 are also observed. Corresponding lines for helium and carbon are Henα, Cnα, and so on.^[25] The discovery of lines withn > 100 was surprising, as even in the very low densities of interstellar space, manyorders of magnitude lower than the best laboratory vacuums attainable on Earth, it had been expected that such highly-excited atoms would be frequently destroyed by collisions, rendering the lines unobservable. Improved theoretical analysis showed that this effect had been overestimated, althoughcollisional broadening does eventually limit detectability of the lines at very highn.^[25] The record wavelength for hydrogen is λ = 73 cm for H253α, implying atomic diameters of a few microns, and for carbon, λ = 18  metres, from C732α,^[26] from atoms with a diameter of 57 micron.

RRLs from hydrogen and helium are produced in highly ionized regions (H II regions and theWarm Ionised Medium). Carbon has a lowerionization energy than hydrogen, and so singly-ionized carbon atoms, and the corresponding recombining Rydberg states, exist further from the ionizing stars, in so-called C II regions which form thick shells around H II regions. The larger volume partially compensates for the low abundance of C compared to H, making the carbon RRLs detectable.

In the absence of collisional broadening, the wavelengths of RRLs are modified only by theDoppler effect, so the measured wavelength,${\displaystyle \lambda }$, is usually converted to radial velocity,${\displaystyle v\approx c(\lambda -{\lambda }_0)/{\lambda }_0}$, where${\displaystyle {\lambda }_0}$ is therest-frame wavelength. H II regions in ourGalaxy can have radial velocities up to ±150 km/s, due to their motion relative to Earth as both orbit the centre of the Galaxy.^[27] These motions are regular enough that${\displaystyle v}$ can be used to estimate the position of the H II region on the line of sight and so its 3D position in the Galaxy. Because all astrophysical Rydberg atoms arehydrogenic, the frequencies of transitions for H, He, and C are given by thesame formula, except for the slightly differentreduced mass of the valence electron for each element. This gives helium and carbon lines apparent Doppler shifts of −100 and −140 km/s, respectively, relative to the corresponding hydrogen line.

RRLs are used to detect ionized gas in distant regions of our Galaxy, and also inexternal galaxies, because the radio photons are not absorbed byinterstellar dust, which blocks photons from the more familiar optical transitions.^[28] They are also used to measure the temperature of the ionized gas, via the ratio of line intensity to the continuumbremsstrahlung emission from theplasma.^[25] Since the temperature of H II regions is regulated by line emission from heavier elements such as C, N, and O, recombination lines also indirectly measure their abundance (metallicity).^[29]

RRLs are spread across theradio spectrum with relatively small intervals in wavelength between them, so they frequently occur in radio spectral observations primarily targeted at other spectral lines. For instance, H166α, H167α, and H168α are very close in wavelength to the21-cm line from neutral hydrogen. This allows radio astronomers to study both the neutral and the ionized interstellar medium from the same set of observations.^[30] Since RRLs are numerous and weak, common practice is to average the velocity spectra of several neighbouring lines, to improve sensitivity.

There are a variety of other potential applications of Rydberg atoms in cosmology and astrophysics.^[31]

Due to their large size, Rydberg atoms can exhibit very largeelectric dipole moments. Calculations usingperturbation theory show that this results in strong interactions between two close Rydberg atoms. Coherent control of these interactions combined with their relatively long lifetime makes them a suitable candidate to realize aquantum computer.^[32] In 2010 two-qubitgates were achieved experimentally.^[33][34] Strongly interacting Rydberg atoms also featurequantum critical behavior, which makes them interesting to study on their own.^[35]

Since 2000's Rydberg atoms research encompasses broadly five directions: sensing,quantum optics,^{[36][37][38][39][40][41]} quantum computation,^{[42][43][44][45]}quantum simulation^{[46][2][47][48]} and Rydberg states of matter.^[49][50] High electric dipole moments between Rydberg atomic states are used for radio frequency andterahertz sensing and imaging,^[51][52] includingnon-demolition measurements of individual microwave photons.^[53]Electromagnetically induced transparency was used in combination with strong interactions between two atoms excited in Rydberg state to provide medium that exhibits strongly nonlinear behaviour at the level of individual optical photons.^[54][55] The tuneable interaction between Rydberg states, enabled also first quantum simulation experiments.^[56][57]

In October 2018, theUnited States Army Research Laboratory publicly discussed efforts to develop a super wideband radio receiver using Rydberg atoms.^[58] In March 2020, the laboratory announced that its scientists analysed the Rydberg sensor's sensitivity to oscillating electric fields over an enormous range of frequencies—from 0 to 10¹² Hertz (the spectrum to 0.3mm wavelength). The Rydberg sensor can reliably detect signals over the entire spectrum and compare favourably with other established electric field sensor technologies, such as electro-optic crystals and dipole antenna-coupled passive electronics.^[59][60]

A simple 1/r potential results in a closedKeplerian elliptical orbit. In the presence of an externalelectric field Rydberg atoms can obtain very largeelectric dipole moments making them extremely susceptible to perturbation by the field.Figure 7 shows how application of an external electric field (known in atomic physics as aStark field) changes the geometry of the potential, dramatically changing the behaviour of the electron. A Coulombic potential does not apply anytorque as the force is alwaysantiparallel to the position vector (always pointing along a line running between the electron and the nucleus):

${\displaystyle |\tau |=|\mathbf{r}\times \mathbf{F}|=|\mathbf{r}||\mathbf{F}|\sin \theta }$,

${\displaystyle \theta =\pi \Rightarrow \tau =0}$.

With the application of a static electric field, the electron feels a continuously changing torque. The resulting trajectory becomes progressively more distorted over time, eventually going through the full range of angular momentum fromL =L_MAX, to a straight lineL = 0, to the initial orbit in the opposite senseL = −L_MAX.^[61]

The time period of the oscillation in angular momentum (the time to complete the trajectory infigure 8), almost exactly matches the quantum mechanically predicted period for the wavefunction to return to its initial state, demonstrating the classical nature of the Rydberg atom.

• Heavy Rydberg system
• Old quantum theory
• Quantum chaos
• Rydberg molecule
• Rydberg polaron

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