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= Rydberg atom =
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Introduction
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A Rydberg atom is an excited atom with one or more electrons that have
a very high principal quantum number. These atoms have a number of
peculiar properties including an exaggerated response to electric and
magnetic fields, long decay periods and electron wavefunctions that
approximate, under some conditions, classical orbits of electrons
about the nuclei. The core electrons shield the outer electron from
the electric field of the nucleus such that, from a distance, the
electric potential looks identical to that experienced by the electron
in a hydrogen atom.
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:
: \mathbf{F}=m\mathbf{a} \Rightarrow { ke^2 \over r^2}={mv^2 \over r}
where 'k' = 1/(4�ε0).
Orbital momentum is quantized in units of 'ħ':
: mvr=n\hbar .
Combining these two equations leads to Bohr's expression for the
orbital radius in terms of the principal quantum number, 'n':
: r={n^2\hbar^2 \over ke^2m}.
It is now apparent why Rydberg atoms have such peculiar properties:
the radius of the orbit scales as 'n'2 (the 'n' = 137 state of
hydrogen has an atomic radius ~1 µm) and the geometric cross-section
as 'n'4. Thus Rydberg atoms are extremely large with loosely bound
valence electrons, easily perturbed or ionized by collisions or
external fields.
Because the binding energy of a Rydberg electron is proportional to
1/'r' and hence falls off like 1/'n'2, the energy level spacing falls
off like 1/'n'3 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
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The existence of the Rydberg series was first demonstrated in 1885
when Johann Balmer discovered a simple empirical formula for the
wavelengths of light associated with transitions in atomic hydrogen.
Three years later, the Swedish physicist Johannes Rydberg presented a
generalized and more intuitive version of Balmer's formula that came
to be known as the Rydberg formula. This formula indicated the
existence of an infinite series of ever more closely spaced discrete
energy levels converging on a finite limit.
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. 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
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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
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Much early experimental work on Rydberg atoms relied on the use of
collimated beams of fast electrons incident on ground-state atoms.
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,
: e^- + A \rarr A^* + e^- .
Because the electron can retain any arbitrary amount of its initial
kinetic energy, this process always results in a population with a
broad spread of different energies.
Charge exchange excitation
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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,
: A^+ + B \rarr 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
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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,
absolutely specifying the 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,
: A + \gamma \rarr 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
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An atom in a Rydberg state has a valence electron in a large orbit far
from the ion core; in such an orbit, the outermost electron feels an
almost hydrogenic, Coulomb potential, 'U'C from a compact ion core
consisting of a nucleus with 'Z' protons and the lower electron shells
filled with 'Z'-1 electrons. An electron in the spherically symmetric
Coulomb potential has potential energy:
:U_\text{C} = -\dfrac{e^2}{4\pi\varepsilon_0r}.
The similarity of the effective potential �seen� by the outer electron
to the hydrogen potential is a defining characteristic of Rydberg
states and explains why the electron wavefunctions approximate to
classical orbits in the limit of the correspondence principle. 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 useful Bohr and Rutherford 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. For an atom in a multiple Rydberg state, the additional
term, 'Uee', includes a summation of each 'pair' of highly excited
electrons:
: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/'r'4
core polarization term in the potential. The interaction between an
induced dipole and the charge that produces it is always attractive so
this contribution is always negative,
:U_\text{pol} = -\dfrac{e^2\alpha_\text{d}}{(4\pi\varepsilon_0)^2r^4},
:where αd 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.
Quantum-mechanical details
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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:
: E_\text{B} = -\dfrac{Ry}{n^2},
where 'Ry' = 13.6 eV is the Rydberg constant. The low binding energy
at high values of 'n' 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 a quantum defect, δ'l', into the expression for the
binding energy:
:E_\text{B} = -\dfrac{Ry}{(n-\delta_l)^2}.
Electron wavefunctions
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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 'ni', energetically
close to the state of the outer electron 'no', then its wavefunction
becomes almost as large as the first (a double Rydberg state). This
occurs as 'ni' approaches 'no' and leads to a condition where the size
of the two electron�s orbits are related; a condition sometimes
referred to as 'radial correlation'. 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. 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.
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
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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,
: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 degeneracy 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 the Inglis-Teller limit,
:F_\text{IT} = \dfrac{e}{12\pi\varepsilon_0a_0^2n^5}.
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.
Precision measurements of trapped Rydberg atoms
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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.
Investigating diamagnetic effects
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The large sizes and low binding energies of Rydberg atoms lead to a
high magnetic susceptibility, \chi. As diamagnetic effects scale with
the area of the orbit and the area is proportional to the radius
squared ('A' � 'n'4), effects impossible to detect in ground state
atoms become obvious in Rydberg atoms, which demonstrate very large
diamagnetic shifts.
Rydberg atoms exhibit strong electric-dipole coupling of the atoms to
electromagnetic fields and has been used to detect radio
communications.
In plasmas
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Rydberg atoms form commonly in plasmas 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 high kinetic energy often form autoionising 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.
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.
In astrophysics
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It has been suggested that Rydberg atoms are common in interstellar
space and could be observed from earth. Since the density within
interstellar gas clouds is many orders of magnitude lower than the
best laboratory vacuums attainable on Earth, Rydberg states could
persist for long periods of time without being destroyed by
collisions.
Strongly interacting systems
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Due to their large size, Rydberg atoms can exhibit very large electric
dipole moments. Calculations using perturbation 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 a quantum
computer. In 2010 two-qubit gates were achieved experimentally.
Strongly interacting Rydberg atoms also feature quantum critical
behavior, which makes them interesting to study on their own.
Current research directions
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Since 2000's Rydberg atoms research encompasses broadly three
directions: sensing, quantum optics and quantum simulation . High
electric dipole moments between Rydberg atomic states are used for
radiofrequency and terahertz sensing and imaging, including
non-demolition measurements of individual microwave photons.
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 . The tuneable interaction between
Rydberg states, enabled also first quantum simulation experiments.
Classical simulation
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A simple 1/'r' potential results in a closed Keplerian elliptical
orbit. In the presence of an external electric field Rydberg atoms
can obtain very large electric 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
a Stark field) changes the geometry of the potential, dramatically
changing the behaviour of the electron. A Coulombic potential does
not apply any torque as the force is always antiparallel to the
position vector (always pointing along a line running between the
electron and the nucleus):
:|\mathbf{\tau}|=|\mathbf{r} \times
\mathbf{F}|=|\mathbf{r}||\mathbf{F}|\sin\theta ,
:\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.
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
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* Heavy Rydberg system
* Old quantum theory
* Quantum chaos
* Rydberg molecule
* Rydberg polaron
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Original Article:
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