Spontaneous Emission
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Spontaneous emission is the process in which a quantum mechanical system (such as a
molecule A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...
, an
atom Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...
or a
subatomic particle In physics, a subatomic particle is a particle smaller than 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 baryon, lik ...
) transits from an excited energy state to a lower energy state (e.g., its 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 particles that can ...
. 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 a spontaneous emission of radiation from an electronically or vibrationally excited species not in thermal equilibrium with its environment. A luminescent object emits ''cold light'' in contrast to incandescence, where an obje ...
. 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 emission ...
, chemiluminescence etc.). If the excitation is affected by the absorption of radiation the spontaneous emission is called
fluorescence Fluorescence is one of two kinds of photoluminescence, the emission of light by a substance that has absorbed light or other electromagnetic radiation. When exposed to ultraviolet radiation, many substances will glow (fluoresce) with colore ...
. 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 fluor ...
. 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'' originated as an acronym for light amplification by stimulated emission of radi ...
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 ...
. Spontaneous emission cannot be explained by classical electromagnetic theory and is fundamentally a quantum process. The first person to correctly predict the phenomenon of spontaneous emission was
Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
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 Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
and
quantum optics Quantum optics is a branch of atomic, molecular, and optical physics and quantum chemistry that studies the behavior of photons (individual quanta of light). It includes the study of the particle-like properties of photons and their interaction ...
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 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 Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
. 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 In quantum optics, the Jaynes–Cummings model (sometimes abbreviated JCM) is a theoretical model that describes the system of a Two-level system, two-level atom interacting with a quantized mode of an optical cavity (or a bosonic field), with o ...
was developed describing the system of a two-level atom interacting with a quantized field mode (i.e. the vacuum) within an
optical cavity An optical cavity, resonating cavity or optical resonator is an arrangement of mirrors or other optical elements that confines light waves similarly to how a cavity resonator confines microwaves. Optical cavities are a major component of lasers, ...
. 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 ...
(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 E_2, it may spontaneously decay to a lower lying level (e.g., the ground state) with energy E_1, releasing the difference in energy between the two states as a photon. The photon will have
angular frequency In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine ...
\omega and an
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
\hbar \omega: :E_2 - E_1 = \hbar \omega, where \hbar is the
reduced Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
. Note: \hbar \omega = h\nu, where h is the
Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
and \nu is the linear
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
. The phase 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 ...
. 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 t is given by N(t), the rate at which N decays is: :\frac = -A_ N(t), where A_ is the rate of spontaneous emission. In the rate-equation A_ 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: :N(t) =N(0) e^= N(0) e^, where N(0) is the initial number of light sources in the excited state, t is the time and \Gamma_ is the radiative decay rate of the transition. The number of excited states N 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 conside ...
. After one lifetime, the number of excited states decays to 36.8% of its original value (\frac-time). The radiative decay rate \Gamma_ is inversely proportional to the lifetime \tau_: :A_=\Gamma_=\frac.


Theory

Spontaneous transitions were not explainable within the framework of the
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
, 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 Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, 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 Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
. In quantum electrodynamics (or QED), the electromagnetic field has a 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 fluctuations 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 The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
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: : , \psi(t)\rangle = a(t)e^, e;0\rangle + \sum_ b_(t)e^, g;1_\rangle where , e;0\rangle and a(t) are the atomic excited state-electromagnetic vacuum wavefunction and its probability amplitude, , g;1_\rangle and b_(t) are the ground state atom with a single photon (of mode ks ) wavefunction and its probability amplitude, \omega_0 is the atomic transition frequency, and \omega_k = c, k, is the frequency of the photon. The sum is over k and s , which are the
wavenumber In the physical sciences, the wavenumber (or wave number), also known as repetency, is the spatial frequency of a wave. Ordinary wavenumber is defined as the number of wave cycles divided by length; it is a physical quantity with dimension of ...
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 ( , b(t), ^2), 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 ...
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 Victor Weisskopf and
Eugene Wigner Eugene Paul Wigner (, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of th ...
in 1930 in a landmark paper. The Weisskopf-Wigner calculation remains the standard approach to spontaneous radiation emission in atomic and molecular physics. Dirac had also developed the same calculation a couple of years prior to the paper by Wigner and Weisskopf.


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 a ...
. 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 (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressur ...
, the rate of spontaneous emission in the dipole approximation is given by: : \Gamma_(\omega)= \frac = \frac : \frac = 4 \alpha , \langle 1, \mathbf, 2\rangle , ^2 where \omega is the emission frequency, n is the index of refraction, \mu_ is the transition dipole moment, \varepsilon_0 is the vacuum permittivity, \hbar is the
reduced Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
, c is the vacuum
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
, and \alpha is the
fine-structure constant In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Alpha, Greek letter ''alpha''), is a Dimensionless physical constant, fundamental physical constant that quantifies the strength of the el ...
. The expression , \langle 1, \mathbf, 2\rangle, stands for the definition of the transition dipole moment , \mu_, =, \langle 1, \mathbf, 2\rangle, for dipole moment operator \mathbf=q\mathbf, where q is the elementary charge and \mathbf 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 \omega^3. In contrast with atoms, which have a discrete emission spectrum, quantum dots can be tuned continuously by changing their size. This property has been used to check the \omega^3-frequency dependence of the spontaneous emission rate as described by Fermi's golden rule.


Radiative and nonradiative decay: the quantum efficiency

In the rate-equation above, it is assumed that decay of the number of excited states N 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 \Gamma_, radiative and nonradiative rates should be summed: :\Gamma_=\Gamma_ + \Gamma_ where \Gamma_ is the total decay rate, \Gamma_ is the radiative decay rate and \Gamma_ the nonradiative decay rate. The quantum efficiency (QE) is defined as the fraction of emission processes in which emission of light is involved: : \text=\frac. In nonradiative relaxation, the energy is released as phonons, more commonly known as
heat In thermodynamics, heat is energy in transfer between a thermodynamic system and its surroundings by such mechanisms as thermal conduction, electromagnetic radiation, and friction, which are microscopic in nature, involving sub-atomic, ato ...
. 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 with electrical conductivity between that of a conductor and an insulator. Its conductivity can be modified by adding impurities (" doping") to its crystal structure. When two regions with different doping level ...
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 and solid-state chemistry, a band gap, also called a bandgap or energy gap, is an energy range in a solid where no electronic states exist. In graphs of the electronic band structure of solids, the band gap refers to the ...
in semiconductors. Large nonradiative transitions do not occur frequently because the
crystal structure In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat ...
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'' originated as an acronym for light amplification by stimulated emission of radi ...
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 ...
can be used to boost an optical signal.


Radiative cascade

If emission leaves a system in an excited state, additional transitions can occur, leading to atomic radiative cascade. For example, if calcium atoms in a low-pressure atomic beam are excited by ultraviolet light from the 41S0 ground state to the 61P1 state, they can decay in three steps, first to 61S0 then to 41P1 and finally to the ground state. The photons from the second and third transitions have correlated polarizations demonstrating
quantum entanglement Quantum entanglement is the phenomenon where the quantum state of each Subatomic particle, particle in a group cannot be described independently of the state of the others, even when the particles are separated by a large distance. The topic o ...
. These correlations were used by John Clauser and Alain Aspect in work that contributed to their 2022 Nobel prize in physics.


See also

* Absorption (optics) *
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 ...
*
Emission spectrum The emission spectrum of a chemical element or chemical compound is the Spectrum (physical sciences), spectrum of frequencies of electromagnetic radiation emitted due to electrons making a atomic electron transition, transition from a high energ ...
*
Spectral line A spectral line is a weaker or stronger region in an otherwise uniform and continuous spectrum. It may result from emission (electromagnetic radiation), emission or absorption (electromagnetic radiation), absorption of light in a narrow frequency ...
* Atomic spectral line *
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 popula ...
* Purcell effect * Photonic crystal * Vacuum Rabi oscillation *
Jaynes–Cummings model In quantum optics, the Jaynes–Cummings model (sometimes abbreviated JCM) is a theoretical model that describes the system of a Two-level system, two-level atom interacting with a quantized mode of an optical cavity (or a bosonic field), with o ...


References


External links


Detail calculation of the Spontaneous Emission: Weisskopf-Wigner Theory


{{DEFAULTSORT:Spontaneous Emission Physical phenomena Laser science Electromagnetic radiation Charge carriers