Rydberg atom

A Rydberg atom is an

In spite of its shortcomings, the Bohr model of the atom is useful in explaining these properties. Classically, an electron in a circular orbit of radius r, about a hydrogen nucleus of charge +e, obeys Newton's second law:

${\displaystyle \mathbf {F} =m\mathbf {a} \Rightarrow {ke^{2} \over r^{2}}={mv^{2} \over r}}$

where k = 1/(4π

Orbital momentum is

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

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

${\displaystyle r={n^{2}\hbar ^{2} \over ke^{2}m}.}$

It is now apparent why Rydberg atoms have such peculiar properties: the radius of the orbit scales as n² (the n = 137 state of hydrogen has an atomic radius ~1 µm) and the geometric cross-section as n⁴. Thus, Rydberg atoms are extremely large, with loosely bound

Because the binding 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 first ionization energy. These closely spaced Rydberg states form what is commonly referred to as the Rydberg series. Figure 2 shows some of the energy levels of the lowest three values of orbital angular momentum in lithium.

History

The existence of the Rydberg series was first demonstrated in 1885 when

This series was qualitatively explained in 1913 by Niels Bohr with his semiclassical model of the hydrogen atom in which quantized values of angular momentum lead to the observed discrete energy levels.^[7][8] A full quantitative derivation of the observed spectrum was derived by Wolfgang Pauli in 1926 following development of quantum mechanics by Werner Heisenberg and others.

Methods of production

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

Electron impact excitation

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 electron kinetic energy to increase the atoms' internal energy exciting to a broad range of different states including many high-lying Rydberg states,

${\displaystyle e^{-}+A\rightarrow A^{*}+e^{-}}$.

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.

Charge exchange excitation

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

${\displaystyle A^{+}+B\rightarrow A^{*}+B^{+}}$.

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.

Optical excitation

The arrival of tunable dye lasers in the 1970s allowed a much greater level of control over populations of excited atoms. In optical excitation, the incident photon 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,

${\displaystyle A+\gamma \rightarrow A^{*}}$.

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

For atoms with a large valence electron binding energy (equivalent to a large first ionization 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.

Hydrogenic potential

An atom in a

• 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 each pair of highly excited electrons:

${\displaystyle U_{ee}={\dfrac {e^{2}}{4\pi \varepsilon _{0}}}\sum _{i<j}{\dfrac {1}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}}$.

• If the valence electron has very low angular momentum (interpreted classically as an extremely eccentric 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 an induced dipole and the charge that produces it is always attractive so this contribution is always negative,

${\displaystyle U_{\text{pol}}=-{\dfrac {e^{2}\alpha _{\text{d}}}{(4\pi \varepsilon _{0})^{2}r^{4}}}}$,

where α_d is the dipole polarizability. 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-mechanical details

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

${\displaystyle E_{\text{B}}=-{\frac {\rm {Ry}}{n^{2}}}}$,

where Ry = 13.6

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

${\displaystyle E_{\text{B}}=-{\frac {\rm {Ry}}{(n-\delta _{l})^{2}}}}$.

Electron wavefunctions

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 high l 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 atomic Hamiltonian:

• If a second electron is excited into a state n_i, energetically close to the state of the outer electron n_o, then its wavefunction becomes almost as large as the first (a double Rydberg state). This occurs as n_i approaches n_o and leads to a condition where the size of the two electron’s orbits are related;^[12] a condition sometimes referred to as radial correlation.^[1] An electron-electron repulsion term must be included in the atomic Hamiltonian.
• Polarization of the ion core produces an anisotropic potential that causes an angular correlation between the motions of the two outermost electrons.^[1][15] This can be thought of as a tidal 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 momentum l, is periodically localised within the shells of inner electrons and interacts with the full charge of the nucleus.^[14] Figure 4 shows a semi-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.

In external fields

Figure 5. Computed energy level spectra of hydrogen in an electric field near n=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 adjacent n-manifolds cross at the Inglis–Teller limit.

Figure 6. Computed energy level spectra of lithium in an electric field near n=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 hence avoided crossings of the energy levels.

The large separation between the electron and ion-core in a Rydberg atom makes possible an extremely large electric dipole moment, d. There is an energy associated with the presence of an electric dipole in an electric field, F, known in atomic physics as a Stark 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 adjacent n-levels in the Rydberg series means that states can approach

In the hydrogen atom, the pure 1/r Coulomb potential does not couple Stark states from adjacent n-manifolds resulting in real crossings as shown in figure 5. The presence of additional terms in the potential energy can lead to coupling resulting in avoided crossings as shown for lithium in figure 6.

Applications and further research

Precision measurements of trapped Rydberg atoms

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]

Investigating diamagnetic effects

The large sizes and low binding energies of Rydberg atoms lead to a high magnetic 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]

In plasmas

Rydberg atoms form commonly in

Condensation of Rydberg atoms forms Rydberg 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]

In astrophysics (Radio recombination lines)

Rydberg atoms occur in space due to the dynamic equilibrium between

In the absence of collisional broadening, the wavelengths of RRLs are modified only by the Doppler 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 the rest-frame wavelength. H II regions in our Galaxy 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 are

RRLs are used to detect ionized gas in distant regions of our Galaxy, and also in external galaxies, because the radio photons are not absorbed by interstellar 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 continuum bremsstrahlung emission from the plasma.^[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 the

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

Strongly interacting systems

Due to their large size, Rydberg atoms can exhibit very large

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 quantum matters.^[49][50] High electric dipole moments between Rydberg atomic states are used for radio frequency and terahertz sensing and imaging,^[51][52] including non-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, the United 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]

Classical simulation

A simple 1/r potential results in a closed

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

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

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 from L = L_MAX, to a straight line L=0, to the initial orbit in the opposite sense L = -L_MAX.^[61]

The time period of the oscillation in angular momentum (the time to complete the trajectory in figure 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.

See also

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

References

Wikimedia Commons has media related to Rydberg atoms.