Spontaneous emission is the process in which a
quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
system (such as a
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
, an
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
or a
subatomic particle
In physical sciences, a subatomic particle is a particle that composes an atom. According to the Standard Model of particle physics, a subatomic particle can be either a composite particle, which is composed of other particles (for example, a pr ...
) transits from an
excited energy state to a lower energy state (e.g., its
ground state
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
) and emits a quantized amount of energy in the form of a
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
. Spontaneous emission is ultimately responsible for most of the light we see all around us; it is so ubiquitous that there are many names given to what is essentially the same process. If atoms (or molecules) are excited by some means other than heating, the spontaneous emission is called
luminescence
Luminescence is spontaneous emission of light by a substance not resulting from heat; or "cold light".
It is thus a form of cold-body radiation. It can be caused by chemical reactions, electrical energy, subatomic motions or stress on a cryst ...
. For example, fireflies are luminescent. And there are different forms of luminescence depending on how excited atoms are produced (
electroluminescence
Electroluminescence (EL) is an optical phenomenon, optical and electrical phenomenon, in which a material emits light in response to the passage of an electric current or to a strong electric field. This is distinct from black body light emissi ...
,
chemiluminescence
Chemiluminescence (also chemoluminescence) is the emission of light (luminescence) as the result of a chemical reaction. There may also be limited emission of heat. Given reactants A and B, with an excited intermediate ◊,
: + -> lozenge -> ...
etc.). If the excitation is affected by the absorption of radiation the spontaneous emission is called
fluorescence
Fluorescence is the emission of light by a substance that has absorbed light or other electromagnetic radiation. It is a form of luminescence. In most cases, the emitted light has a longer wavelength, and therefore a lower photon energy, tha ...
. Sometimes molecules have a metastable level and continue to fluoresce long after the exciting radiation is turned off; this is called
phosphorescence
Phosphorescence is a type of photoluminescence related to fluorescence. When exposed to light (radiation) of a shorter wavelength, a phosphorescent substance will glow, absorbing the light and reemitting it at a longer wavelength. Unlike fluo ...
. Figurines that glow in the dark are phosphorescent.
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
s start via spontaneous emission, then during continuous operation work by
stimulated emission
Stimulated emission is the process by which an incoming photon of a specific frequency can interact with an excited atomic electron (or other excited molecular state), causing it to drop to a lower energy level. The liberated energy transfers to th ...
.
Spontaneous emission cannot be explained by
classical electromagnetic theory and is fundamentally a quantum process. According to the American Physical Society, the first person to correctly predict the phenomenon of spontaneous emission was
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
in a series of papers starting in 1916, culminating in what is now called the
Einstein A Coefficient.
Einstein's quantum theory of radiation anticipated ideas later expressed in
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
and
quantum optics
Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have b ...
by several decades.
Later, after the formal discovery of quantum mechanics in 1926, the rate of spontaneous emission was accurately described from first principles by
Dirac
Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety o ...
in his quantum theory of radiation,
the precursor to the theory which he later called
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
.
Contemporary physicists, when asked to give a physical explanation for spontaneous emission, generally invoke the
zero-point energy
Zero-point energy (ZPE) is the lowest possible energy that a quantum mechanical system may have. Unlike in classical mechanics, quantum systems constantly Quantum fluctuation, fluctuate in their lowest energy state as described by the Heisen ...
of the electromagnetic field. In 1963, the
Jaynes–Cummings model
The Jaynes–Cummings model (sometimes abbreviated JCM) is a theoretical model in quantum optics. It describes the system of a two-level atom interacting with a quantized mode of an optical cavity (or a bosonic field), with or without the prese ...
was developed describing the system of a
two-level atom interacting with a quantized field mode (i.e. the vacuum) within an optical cavity. It gave the nonintuitive prediction that the rate of spontaneous emission could be controlled depending on the boundary conditions of the surrounding vacuum field. These experiments gave rise to
cavity quantum electrodynamics Cavity quantum electrodynamics (cavity QED) is the study of the interaction between light confined in a reflective cavity and atoms or other particles, under conditions where the quantum nature of photons is significant. It could in principle be u ...
(CQED), the study of effects of mirrors and cavities on radiative corrections.
Introduction
If a light source ('the atom') is in an excited state with energy
, it may spontaneously decay to a lower lying level (e.g., the ground state) with energy
, releasing the difference in energy between the two states as a photon. The photon will have
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
and an
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
:
:
where
is the
reduced Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
. Note:
, where
is the
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
and
is the linear
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
. The
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
of the photon in spontaneous emission is random as is the direction in which the photon propagates. This is not true for
stimulated emission
Stimulated emission is the process by which an incoming photon of a specific frequency can interact with an excited atomic electron (or other excited molecular state), causing it to drop to a lower energy level. The liberated energy transfers to th ...
. An energy level diagram illustrating the process of spontaneous emission is shown below:
If the number of light sources in the excited state at time
is given by
, the rate at which
decays is:
:
where
is the rate of spontaneous emission. In the rate-equation
is a proportionality constant for this particular transition in this particular light source. The constant is referred to as the ''
Einstein A coefficient'', and has units s.
The above equation can be solved to give:
:
where
is the initial number of light sources in the excited state,
is the time and
is the radiative decay rate of the transition. The number of excited states
thus decays exponentially with time, similar to
radioactive decay
Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is consid ...
. After one lifetime, the number of excited states decays to 36.8% of its original value (
-time). The radiative decay rate
is inversely proportional to the lifetime
:
:
Theory
Spontaneous transitions were not explainable within the framework of the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
, in which the electronic energy levels were quantized, but the electromagnetic field was not. Given that the eigenstates of an atom are properly diagonalized, the overlap of the wavefunctions between the excited state and the ground state of the atom is zero. Thus, in the absence of a quantized electromagnetic field, the excited state atom cannot decay to the ground state. In order to explain spontaneous transitions, quantum mechanics must be extended to a
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, wherein the electromagnetic field is quantized at every point in space. The quantum field theory of electrons and electromagnetic fields is known as
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
.
In quantum electrodynamics (or QED), the electromagnetic field has a
ground state
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
, the
QED vacuum, which can mix with the excited stationary states of the atom.
As a result of this interaction, the "stationary state" of the atom is no longer a true
eigenstate of the combined system of the atom plus electromagnetic field. In particular, the electron transition from the excited state to the electronic ground state mixes with the transition of the electromagnetic field from the ground state to an excited state, a field state with one photon in it. Spontaneous emission in free space depends upon
vacuum fluctuation
In quantum physics, a quantum fluctuation (also known as a vacuum state fluctuation or vacuum fluctuation) is the temporary random change in the amount of energy in a point in space, as prescribed by Werner Heisenberg's uncertainty principle. ...
s to get started.
[
][
]
Although there is only one electronic transition from the excited state to ground state, there are many ways in which the electromagnetic field may go from the ground state to a one-photon state. That is, the electromagnetic field has infinitely more degrees of freedom, corresponding to the different directions in which the photon can be emitted. Equivalently, one might say that the
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
offered by the electromagnetic field is infinitely larger than that offered by the atom. This infinite degree of freedom for the emission of the photon results in the apparent irreversible decay, i.e., spontaneous emission.
In the presence of electromagnetic vacuum modes, the combined atom-vacuum system is explained by the superposition of the wavefunctions of the excited state atom with no photon and the ground state atom with a single emitted photon:
:
where
and
are the atomic excited state-electromagnetic vacuum wavefunction and its probability amplitude,
and
are the ground state atom with a single photon (of mode
) wavefunction and its probability amplitude,
is the atomic transition frequency, and
is the frequency of the photon. The sum is over
and
, which are the wavenumber and polarization of the emitted photon, respectively. As mentioned above, the emitted photon has a chance to be emitted with different wavenumbers and polarizations, and the resulting wavefunction is a superposition of these possibilities. To calculate the probability of the atom at the ground state (
), one needs to solve the time evolution of the wavefunction with an appropriate Hamiltonian.
To solve for the transition amplitude, one needs to average over (integrate over) all the vacuum modes, since one must consider the probabilities that the emitted photon occupies various parts of phase space equally. The "spontaneously" emitted photon has infinite different modes to propagate into, thus the probability of the atom re-absorbing the photon and returning to the original state is negligible, making the atomic decay practically irreversible. Such irreversible time evolution of the atom-vacuum system is responsible for the apparent spontaneous decay of an excited atom. If one were to keep track of all the vacuum modes, the combined atom-vacuum system would undergo unitary time evolution, making the decay process reversible.
Cavity quantum electrodynamics Cavity quantum electrodynamics (cavity QED) is the study of the interaction between light confined in a reflective cavity and atoms or other particles, under conditions where the quantum nature of photons is significant. It could in principle be u ...
is one such system where the vacuum modes are modified resulting in the reversible decay process, see also
Quantum revival. The theory of the spontaneous emission under the QED framework was first calculated by Weisskopf and Wigner.
Rate of spontaneous emission
The rate of spontaneous emission (i.e., the radiative rate) can be described by
Fermi's golden rule
In quantum physics, Fermi's golden rule is a formula that describes the transition rate (the probability of a transition per unit time) from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of ...
. The rate of emission depends on two factors: an 'atomic part', which describes
the internal structure of the light source and a 'field part', which describes the density of electromagnetic modes of the environment. The atomic part describes the strength of a transition between two states in terms of transition moments. In a homogeneous medium, such as
free space
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
, the rate of spontaneous emission in the dipole approximation is given by:
:
:
where
is the emission frequency,
is the
index of refraction
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
,
is the
transition dipole moment
The transition dipole moment or transition moment, usually denoted \mathbf_ for a transition between an initial state, m, and a final state, n, is the electric dipole moment associated with the transition between the two states. In general the tra ...
,
is the
vacuum permittivity
Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric consta ...
,
is the
reduced Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
,
is the vacuum
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
, and
is the
fine-structure constant
In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between ele ...
. The expression
stands for the definition of the transition dipole moment
for dipole moment operator
, where
is the elementary charge and
stands for position operator. (This approximation breaks down in the case of inner shell electrons in high-Z atoms.) The above equation clearly shows that the rate of spontaneous emission in free space increases proportionally to
.
In contrast with atoms, which have a discrete emission spectrum,
quantum dots
Quantum dots (QDs) are semiconductor particles a few nanometres in size, having optical and electronic properties that differ from those of larger particles as a result of quantum mechanics. They are a central topic in nanotechnology. When the ...
can be tuned continuously by changing their size. This property has been used to check the
-frequency dependence of the spontaneous emission rate as described by Fermi's golden rule.
[A. F. van Driel, G. Allan, C. Delerue, P. Lodahl,W. L. Vos and D. Vanmaekelbergh,
Frequency-dependent spontaneous emission rate from CdSe
and CdTe nanocrystals: Influence of dark states, Physical Review Letters,
95, 236804 (2005]
Phys. Rev. Lett. 95, 236804 (2005) - Frequency-Dependent Spontaneous Emission Rate from CdSe and CdTe Nanocrystals: Influence of Dark States (aps.org)
/ref>
Radiative and nonradiative decay: the quantum efficiency
In the rate-equation above, it is assumed that decay of the number of excited states only occurs under emission of light. In this case one speaks of full radiative decay and this means that the quantum efficiency is 100%. Besides radiative decay, which occurs under the emission of light, there is a second decay mechanism; nonradiative decay. To determine the total decay rate , radiative and nonradiative rates should be summed:
:
where is the total decay rate, is the radiative decay rate and the nonradiative decay rate. The quantum efficiency (QE) is defined as the fraction of emission processes in which emission of light is involved:
:
In nonradiative relaxation, the energy is released as phonon
In physics, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...
s, more commonly known as heat
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
. Nonradiative relaxation occurs when the energy difference between the levels is very small, and these typically occur on a much faster time scale than radiative transitions. For many materials (for instance, semiconductor
A semiconductor is a material which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...
s), electrons move quickly from a high energy level to a meta-stable level via small nonradiative transitions and then make the final move down to the bottom level via an optical or radiative transition. This final transition is the transition over the bandgap
In solid-state physics, a band gap, also called an energy gap, is an energy range in a solid where no electronic states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference (in ...
in semiconductors. Large nonradiative transitions do not occur frequently because the crystal structure
In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystal, crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric pat ...
generally cannot support large vibrations without destroying bonds (which generally doesn't happen for relaxation). Meta-stable states form a very important feature that is exploited in the construction of laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
s. Specifically, since electrons decay slowly from them, they can be deliberately piled up in this state without too much loss and then stimulated emission
Stimulated emission is the process by which an incoming photon of a specific frequency can interact with an excited atomic electron (or other excited molecular state), causing it to drop to a lower energy level. The liberated energy transfers to th ...
can be used to boost an optical signal.
See also
* Absorption (optics)
In physics, absorption of electromagnetic radiation is how matter (typically electrons bound in atoms) takes up a photon's energy — and so transforms electromagnetic energy into internal energy of the absorber (for example, thermal energy). A ...
* Stimulated emission
Stimulated emission is the process by which an incoming photon of a specific frequency can interact with an excited atomic electron (or other excited molecular state), causing it to drop to a lower energy level. The liberated energy transfers to th ...
* Emission spectrum
The emission spectrum of a chemical element or chemical compound is the spectrum of frequencies of electromagnetic radiation emitted due to an electron making a atomic electron transition, transition from a high energy state to a lower energy st ...
* Spectral line
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to iden ...
* Atomic spectral line
Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter w ...
* Laser science
Laser science or laser physics is a branch of optics that describes the theory and practice of lasers.
Laser science is principally concerned with quantum electronics, laser construction, optical cavity design, the physics of producing a popul ...
* Purcell effect
Henry Purcell (, rare: September 1659 – 21 November 1695) was an English composer.
Purcell's style of Baroque music was uniquely English, although it incorporated Italian and French elements. Generally considered among the greatest Eng ...
* Photonic crystal
A photonic crystal is an optical nanostructure in which the refractive index changes periodically. This affects the propagation of light in the same way that the structure of Crystal structure, natural crystals gives rise to X-ray crystallograp ...
* Vacuum Rabi oscillation
* Jaynes–Cummings model
The Jaynes–Cummings model (sometimes abbreviated JCM) is a theoretical model in quantum optics. It describes the system of a two-level atom interacting with a quantized mode of an optical cavity (or a bosonic field), with or without the prese ...
References
External links
Detail calculation of the Spontaneous Emission: Weisskopf-Wigner Theory
{{DEFAULTSORT:Spontaneous Emission
Physical phenomena
Laser science
Electromagnetic radiation
Charge carriers