The photon is a type of elementary particle,
the quantum of the electromagnetic field including
electromagnetic radiation such as light, and
the force carrier for the electromagnetic
force (even when static via virtual particles).
Invariant mass of the photon is zero; it always
moves at the speed of light within a vacuum.
Like all elementary particles, photons are
currently best explained by quantum mechanics
and exhibit wave–particle duality, exhibiting
properties of both waves and particles. For
example, a single photon may be refracted
by a lens and exhibit wave interference with
itself, and it can behave as a particle with
definite and finite measurable position or
momentum, though not both at the same time
as per Heisenberg's uncertainty principle.
The photon's wave and quantum qualities are
two observable aspects of a single phenomenon—they
cannot be described by any mechanical model;
a representation of this dual property of
light that assumes certain points on the wavefront
to be the seat of the energy is not possible.
The quanta in a light wave are not spatially
localized.
The modern concept of the photon was developed
gradually by Albert Einstein in the early
20th century to explain experimental observations
that did not fit the classical wave model
of light. The benefit of the photon model
is that it accounts for the frequency dependence
of light's energy, and explains the ability
of matter and electromagnetic radiation to
be in thermal equilibrium. The photon model
accounts for anomalous observations, including
the properties of black-body radiation, that
others (notably Max Planck) had tried to explain
using semiclassical models. In that model,
light is described by Maxwell's equations,
but material objects emit and absorb light
in quantized amounts (i.e., they change energy
only by certain particular discrete amounts).
Although these semiclassical models contributed
to the development of quantum mechanics, many
further experiments beginning with the phenomenon
of Compton scattering of single photons by
electrons, validated Einstein's hypothesis
that light itself is quantized. In December
1926, American physical chemist Gilbert N.
Lewis coined the widely-adopted name "photon"
for these particles in a letter to Nature.
After Arthur H. Compton won the Nobel Prize
in 1927 for his scattering studies, most scientists
accepted that light quanta have an independent
existence, and the term "photon" was accepted.
In the Standard Model of particle physics,
photons and other elementary particles are
described as a necessary consequence of physical
laws having a certain symmetry at every point
in spacetime. The intrinsic properties of
particles, such as charge, mass, and spin,
are determined by this gauge symmetry. The
photon concept has led to momentous advances
in experimental and theoretical physics, including
lasers, Bose–Einstein condensation, quantum
field theory, and the probabilistic interpretation
of quantum mechanics. It has been applied
to photochemistry, high-resolution microscopy,
and measurements of molecular distances. Recently,
photons have been studied as elements of quantum
computers, and for applications in optical
imaging and optical communication such as
quantum cryptography.
== Nomenclature ==
The word quanta (singular quantum, Latin for
how much) was used before 1900 to mean particles
or amounts of different quantities, including
electricity. In 1900, the German physicist
Max Planck was studying black-body radiation,
and specifically the Ultraviolet Catastrophe:
he suggested that the experimental observations
would be explained if the energy carried by
electromagnetic waves could only be released
in "packets" of energy. In his 1901 article
in Annalen der Physik he called these packets
"energy elements". In 1905, Albert Einstein
published a paper in which he proposed that
many light-related phenomena—including black-body
radiation and the photoelectric effect—would
be better explained by modelling electromagnetic
waves as consisting of spatially localized,
discrete wave-packets. He called such a wave-packet
the light quantum (German: das Lichtquant).The
name photon derives from the Greek word for
light, φῶς (transliterated phôs). Arthur
Compton used photon in 1928, referring to
Gilbert N. Lewis, who coined the term in a
letter to Nature on December 18, 1926. In
fact, the same name was used earlier but was
never widely adopted before Lewis: in 1916
by the American physicist and psychologist
Leonard T. Troland, in 1921 by the Irish physicist
John Joly, in 1924 by the French physiologist
René Wurmser (1890–1993), and in 1926 by
the French physicist Frithiof Wolfers (1891–1971).
The name was suggested initially as a unit
related to the illumination of the eye and
the resulting sensation of light and was used
later in a physiological context. Although
Wolfers's and Lewis's theories were contradicted
by many experiments and never accepted, the
new name was adopted very soon by most physicists
after Compton used it.In physics, a photon
is usually denoted by the symbol γ (the Greek
letter gamma). This symbol for the photon
probably derives from gamma rays, which were
discovered in 1900 by Paul Villard, named
by Ernest Rutherford in 1903, and shown to
be a form of electromagnetic radiation in
1914 by Rutherford and Edward Andrade. In
chemistry and optical engineering, photons
are usually symbolized by hν, which is the
photon energy, where h is Planck constant
and the Greek letter ν (nu) is the photon's
frequency. Much less commonly, the photon
can be symbolized by hf, where its frequency
is denoted by f.
== Physical properties ==
A photon is massless, has no electric charge,
and is a stable particle. A photon has two
possible polarization states. In the momentum
representation of the photon, which is preferred
in quantum field theory, a photon is described
by its wave vector, which determines its wavelength
λ and its direction of propagation. A photon's
wave vector may not be zero and can be represented
either as a spatial 3-vector or as a (relativistic)
four-vector; in the latter case it belongs
to the light cone (pictured). Different signs
of the four-vector denote different circular
polarizations, but in the 3-vector representation
one should account for the polarization state
separately; it actually is a spin quantum
number. In both cases the space of possible
wave vectors is three-dimensional.
The photon is the gauge boson for electromagnetism,
and therefore all other quantum numbers of
the photon (such as lepton number, baryon
number, and flavour quantum numbers) are zero.
Also, the photon does not obey the Pauli exclusion
principle, but instead obeys Bose–Einstein
statistics.
Photons are emitted in many natural processes.
For example, when a charge is accelerated
it emits synchrotron radiation. During a molecular,
atomic or nuclear transition to a lower energy
level, photons of various energy will be emitted,
ranging from radio waves to gamma rays. Photons
can also be emitted when a particle and its
corresponding antiparticle are annihilated
(for example, electron–positron annihilation).In
empty space, the photon moves at c (the speed
of light) and its energy and momentum are
related by E = pc, where p is the magnitude
of the momentum vector p. This derives from
the following relativistic relation, with
m = 0:
E
2
=
p
2
c
2
+
m
2
c
4
.
{\displaystyle E^{2}=p^{2}c^{2}+m^{2}c^{4}.}
The energy and momentum of a photon depend
only on its frequency (ν) or inversely, its
wavelength (λ):
E
=
ℏ
ω
=
h
ν
=
h
c
λ
{\displaystyle E=\hbar \omega =h\nu ={\frac
{hc}{\lambda }}}
p
=
ℏ
k
,
{\displaystyle {\boldsymbol {p}}=\hbar {\boldsymbol
{k}},}
where k is the wave vector (where the wave
number 
k = |k| = 2π/λ), ω = 2πν is the angular
frequency, and ħ = h/2π is the reduced Planck
constant.Since p points in the direction of
the photon's propagation, the magnitude of
the momentum is
p
=
ℏ
k
=
h
ν
c
=
h
λ
.
{\displaystyle p=\hbar k={\frac {h\nu }{c}}={\frac
{h}{\lambda }}.}
The photon also carries a quantity called
spin angular momentum that does not depend
on its frequency. The magnitude of its spin
is √2ħ and the component measured along
its direction of motion, its helicity, must
be ±ħ. These two possible helicities, called
right-handed and left-handed, correspond to
the two possible circular polarization states
of the photon.To illustrate the significance
of these formulae, the annihilation of a particle
with its antiparticle in free space must result
in the creation of at least two photons for
the following reason. In the center of momentum
frame, the colliding antiparticles have no
net momentum, whereas a single photon always
has momentum (since, as we have seen, it is
determined by the photon's frequency or wavelength,
which cannot be zero). Hence, conservation
of momentum (or equivalently, translational
invariance) requires that at least two photons
are created, with zero net momentum. (However,
it is possible if the system interacts with
another particle or field for the annihilation
to produce one photon, as when a positron
annihilates with a bound atomic electron,
it is possible for only one photon to be emitted,
as the nuclear Coulomb field breaks translational
symmetry.) The energy of the two photons,
or, equivalently, their frequency, may be
determined from conservation of four-momentum.
Seen another way, the photon can be considered
as its own antiparticle. The reverse process,
pair production, is the dominant mechanism
by which high-energy photons such as gamma
rays lose energy while passing through matter.
That process is the reverse of "annihilation
to one photon" allowed in the electric field
of an atomic nucleus.
The classical formulae for the energy and
momentum of electromagnetic radiation can
be re-expressed in terms of photon events.
For example, the pressure of electromagnetic
radiation on an object derives from the transfer
of photon momentum per unit time and unit
area to that object, since pressure is force
per unit area and force is the change in momentum
per unit time.Each photon carries two distinct
and independent forms of angular momentum
of light. The spin angular momentum of light
of a particular photon is always either +ħ
or −ħ.
The light orbital angular momentum of a particular
photon can be any integer N, including zero.
=== Experimental checks on photon mass ===
Current commonly accepted physical theories
imply or assume the photon to be strictly
massless. If the photon is not a strictly
massless particle, it would not move at the
exact speed of light, c, in vacuum. Its speed
would be lower and depend on its frequency.
Relativity would be unaffected by this; the
so-called speed of light, c, would then not
be the actual speed at which light moves,
but a constant of nature which is the upper
bound on speed that any object could theoretically
attain in spacetime. Thus, it would still
be the speed of spacetime ripples (gravitational
waves and gravitons), but it would not be
the speed of photons.
If a photon did have non-zero mass, there
would be other effects as well. Coulomb's
law would be modified and the electromagnetic
field would have an extra physical degree
of freedom. These effects yield more sensitive
experimental probes of the photon mass than
the frequency dependence of the speed of light.
If Coulomb's law is not exactly valid, then
that would allow the presence of an electric
field to exist within a hollow conductor when
it is subjected to an external electric field.
This thus allows one to test Coulomb's law
to very high precision. A null result of such
an experiment has set a limit of m ≲ 10−14
eV/c2.Sharper upper limits on the speed of
light have been obtained in experiments designed
to detect effects caused by the galactic vector
potential. Although the galactic vector potential
is very large because the galactic magnetic
field exists on very great length scales,
only the magnetic field would be observable
if the photon is massless. In the case that
the photon has mass, the mass term 1/2m2AμAμ
would affect the galactic plasma. The fact
that no such effects are seen implies an upper
bound on the photon mass of m < 3×10−27
eV/c2. The galactic vector potential can also
be probed directly by measuring the torque
exerted on a magnetized ring. Such methods
were used to obtain the sharper upper limit
of 1.07×10−27 eV/c2 (the equivalent of
10−18 atomic mass units) given by the Particle
Data Group.These sharp limits from the non-observation
of the effects caused by the galactic vector
potential have been shown to be model-dependent.
If the photon mass is generated via the Higgs
mechanism then the upper limit of m ≲ 10−14
eV/c2 from the test of Coulomb's law is valid.
Photons inside superconductors develop a nonzero
effective rest mass; as a result, electromagnetic
forces become short-range inside superconductors.
== Historical development ==
In most theories up to the eighteenth century,
light was pictured as being made up of particles.
Since particle models cannot easily account
for the refraction, diffraction and birefringence
of light, wave theories of light were proposed
by René Descartes (1637), Robert Hooke (1665),
and Christiaan Huygens (1678); however, particle
models remained dominant, chiefly due to the
influence of Isaac Newton. In the early nineteenth
century, Thomas Young and August Fresnel clearly
demonstrated the interference and diffraction
of light and by 1850 wave models were generally
accepted. In 1865, James Clerk Maxwell's prediction
that light was an electromagnetic wave—which
was confirmed experimentally in 1888 by Heinrich
Hertz's detection of radio waves—seemed
to be the final blow to particle models of
light.
The Maxwell wave theory, however, does not
account for all properties of light. The Maxwell
theory predicts that the energy of a light
wave depends only on its intensity, not on
its frequency; nevertheless, several independent
types of experiments show that the energy
imparted by light to atoms depends only on
the light's frequency, not on its intensity.
For example, some chemical reactions are provoked
only by light of frequency higher than a certain
threshold; light of frequency lower than the
threshold, no matter how intense, does not
initiate the reaction. Similarly, electrons
can be ejected from a metal plate by shining
light of sufficiently high frequency on it
(the photoelectric effect); the energy of
the ejected electron is related only to the
light's frequency, not to its intensity.At
the same time, investigations of blackbody
radiation carried out over four decades (1860–1900)
by various researchers culminated in Max Planck's
hypothesis that the energy of any system that
absorbs or emits electromagnetic radiation
of frequency ν is an integer multiple of
an energy quantum E = hν. As shown by Albert
Einstein, some form of energy quantization
must be assumed to account for the thermal
equilibrium observed between matter and electromagnetic
radiation; for this explanation of the photoelectric
effect, Einstein received the 1921 Nobel Prize
in physics.Since the Maxwell theory of light
allows for all possible energies of electromagnetic
radiation, most physicists assumed initially
that the energy quantization resulted from
some unknown constraint on the matter that
absorbs or emits the radiation. In 1905, Einstein
was the first to propose that energy quantization
was a property of electromagnetic radiation
itself. Although he accepted the validity
of Maxwell's theory, Einstein pointed out
that many anomalous experiments could be explained
if the energy of a Maxwellian light wave were
localized into point-like quanta that move
independently of one another, even if the
wave itself is spread continuously over space.
In 1909 and 1916, Einstein showed that, if
Planck's law of black-body radiation is accepted,
the energy quanta must also carry momentum
p = h/λ, making them full-fledged particles.
This photon momentum was observed experimentally
by Arthur Compton, for which he received the
Nobel Prize in 1927. The pivotal question
was then: how to unify Maxwell's wave theory
of light with its experimentally observed
particle nature? The answer to this question
occupied Albert Einstein for the rest of his
life, and was solved in quantum electrodynamics
and its successor, the Standard Model (see
§ Second quantization and § The photon as
a gauge boson, below).
== Einstein's light quantum ==
Unlike Planck, Einstein entertained the possibility
that there might be actual physical quanta
of light—what we now call photons. He noticed
that a light quantum with energy proportional
to its frequency would explain a number of
troubling puzzles and paradoxes, including
an unpublished law by Stokes, the ultraviolet
catastrophe, and the photoelectric effect.
Stokes's law said simply that the frequency
of fluorescent light cannot be greater than
the frequency of the light (usually ultraviolet)
inducing it. Einstein eliminated the ultraviolet
catastrophe by imagining a gas of photons
behaving like a gas of electrons that he had
previously considered. He was advised by a
colleague to be careful how he wrote up this
paper, in order to not challenge Planck, a
powerful figure in physics, too directly,
and indeed the warning was justified, as Planck
never forgave him for writing it.
== Early objections ==
Einstein's 1905 predictions were verified
experimentally in several ways in the first
two decades of the 20th century, as recounted
in Robert Millikan's Nobel lecture. However,
before Compton's experiment showed that photons
carried momentum proportional to their wave
number (1922), most physicists were reluctant
to believe that electromagnetic radiation
itself might be particulate. (See, for example,
the Nobel lectures of Wien, Planck and Millikan.)
Instead, there was a widespread belief that
energy quantization resulted from some unknown
constraint on the matter that absorbed or
emitted radiation. Attitudes changed over
time. In part, the change can be traced to
experiments such as Compton scattering, where
it was much more difficult not to ascribe
quantization to light itself to explain the
observed results.Even after Compton's experiment,
Niels Bohr, Hendrik Kramers and John Slater
made one last attempt to preserve the Maxwellian
continuous electromagnetic field model of
light, the so-called BKS model. To account
for the data then available, two drastic hypotheses
had to be made:
Energy and momentum are conserved only on
the average in interactions between matter
and radiation, but not in elementary processes
such as absorption and emission. This allows
one to reconcile the discontinuously changing
energy of the atom (the jump between energy
states) with the continuous release of energy
as radiation.
Causality is abandoned. For example, spontaneous
emissions are merely emissions stimulated
by a "virtual" electromagnetic field.However,
refined Compton experiments showed that energy–momentum
is conserved extraordinarily well in elementary
processes; and also that the jolting of the
electron and the generation of a new photon
in Compton scattering obey causality to within
10 ps. Accordingly, Bohr and his co-workers
gave their model "as honorable a funeral as
possible". Nevertheless, the failures of the
BKS model inspired Werner Heisenberg in his
development of matrix mechanics.A few physicists
persisted in developing semiclassical models
in which electromagnetic radiation is not
quantized, but matter appears to obey the
laws of quantum mechanics. Although the evidence
from chemical and physical experiments for
the existence of photons was overwhelming
by the 1970s, this evidence could not be considered
as absolutely definitive; since it relied
on the interaction of light with matter, and
a sufficiently complete theory of matter could
in principle account for the evidence. Nevertheless,
all semiclassical theories were refuted definitively
in the 1970s and 1980s by photon-correlation
experiments. Hence, Einstein's hypothesis
that quantization is a property of light itself
is considered to be proven.
== Wave–particle duality and uncertainty
principles ==
Photons, like all quantum objects, exhibit
wave-like and particle-like properties. Their
dual wave–particle nature can be difficult
to visualize. The photon displays clearly
wave-like phenomena such as diffraction and
interference on the length scale of its wavelength.
For example, a single photon passing through
a double-slit experiment exhibits interference
phenomena but only if no measure was made
at the slit. A single photon passing through
a double-slit experiment lands on the screen
with a probability distribution given by its
interference pattern determined by Maxwell's
equations. However, experiments confirm that
the photon is not a short pulse of electromagnetic
radiation; it does not spread out as it propagates,
nor does it divide when it encounters a beam
splitter. Rather, the photon seems to be a
point-like particle since it is absorbed or
emitted as a whole by arbitrarily small systems,
systems much smaller than its wavelength,
such as an atomic nucleus (≈10−15 m across)
or even the point-like electron. Nevertheless,
the photon is not a point-like particle whose
trajectory is shaped probabilistically by
the electromagnetic field, as conceived by
Einstein and others; that hypothesis was also
refuted by the photon-correlation experiments
cited above. According to our present understanding,
the electromagnetic field itself is produced
by photons, which in turn result from a local
gauge symmetry and the laws of quantum field
theory (see § Second quantization and § The
photon as a gauge boson below).
A key element of quantum mechanics is Heisenberg's
uncertainty principle, which forbids the simultaneous
measurement of the position and momentum of
a particle along the same direction. Remarkably,
the uncertainty principle for charged, material
particles requires the quantization of light
into photons, and even the frequency dependence
of the photon's energy and momentum.
An elegant illustration of the uncertainty
principle is Heisenberg's thought experiment
for locating an electron with an ideal microscope.
The position of the electron can be determined
to within the resolving power of the microscope,
which is given by a formula from classical
optics
Δ
x
∼
λ
sin
⁡
θ
{\displaystyle \Delta x\sim {\frac {\lambda
}{\sin \theta }}}
where θ is the aperture angle of the microscope
and λ is the wavelength of the light used
to observe the electron. Thus, the position
uncertainty
Δ
x
{\displaystyle \Delta x}
can be made arbitrarily small by reducing
the wavelength λ. Even if the momentum of
the electron is initially known, the light
impinging on the electron will give it a momentum
"kick"
Δ
p
{\displaystyle \Delta p}
of some unknown amount, rendering the momentum
of the electron uncertain. If light were not
quantized into photons, the uncertainty
Δ
p
{\displaystyle \Delta p}
could be made arbitrarily small by reducing
the light's intensity. In that case, since
the wavelength and intensity of light can
be varied independently, one could simultaneously
determine the position and momentum to arbitrarily
high accuracy, violating the uncertainty principle.
By contrast, Einstein's formula for photon
momentum preserves the uncertainty principle;
since the photon is scattered anywhere within
the aperture, the uncertainty of momentum
transferred equals
Δ
p
∼
p
photon
sin
⁡
θ
=
h
λ
sin
⁡
θ
{\displaystyle \Delta p\sim p_{\text{photon}}\sin
\theta ={\frac {h}{\lambda }}\sin \theta }
giving the product
Δ
x
Δ
p
∼
h
{\displaystyle \Delta x\Delta p\,\sim \,h}
, which is Heisenberg's uncertainty principle.
Thus, the entire world is quantized; both
matter and fields must obey a consistent set
of quantum laws, if either one is to be quantized.The
analogous uncertainty principle for photons
forbids the simultaneous measurement of the
number
n
{\displaystyle n}
of photons (see Fock state and the Second
quantization section below) in an electromagnetic
wave and the phase
ϕ
{\displaystyle \phi }
of that wave
Δ
n
Δ
ϕ
>
1
{\displaystyle \Delta n\Delta \phi >1}
See coherent state and squeezed coherent state
for more details.
Both photons and electrons create analogous
interference patterns when passed through
a double-slit experiment. For photons, this
corresponds to the interference of a Maxwell
light wave whereas, for material particles
(electron), this corresponds to the interference
of the Schrödinger wave equation. Although
this similarity might suggest that Maxwell's
equations describing the photon's electromagnetic
wave are simply Schrödinger's equation for
photons, most physicists do not agree. For
one thing, they are mathematically different;
most obviously, Schrödinger's one equation
for the electron solves for a complex field,
whereas Maxwell's four equations solve for
real fields. More generally, the normal concept
of a Schrödinger probability wave function
cannot be applied to photons. As photons are
massless, they cannot be localized without
being destroyed; technically, photons cannot
have a position eigenstate
|
r
⟩
{\displaystyle |\mathbf {r} \rangle }
, and, thus, the normal Heisenberg uncertainty
principle
Δ
x
Δ
p
>
h
/
2
{\displaystyle \Delta x\Delta p>h/2}
does not pertain to photons. A few substitute
wave functions have been suggested for the
photon, but they have not come into general
use. Instead, physicists generally accept
the second-quantized theory of photons described
below, quantum electrodynamics, in which photons
are quantized excitations of electromagnetic
modes.
Another interpretation, that avoids duality,
is the De Broglie–Bohm theory: known also
as the pilot-wave model. In that theory, the
photon is both, wave and particle. "This idea
seems to me so natural and simple, to resolve
the wave-particle dilemma in such a clear
and ordinary way, that it is a great mystery
to me that it was so generally ignored", J.S.
Bell.
== Bose–Einstein model of a photon gas ==
In 1924, Satyendra Nath Bose derived Planck's
law of black-body radiation without using
any electromagnetism, but rather by using
a modification of coarse-grained counting
of phase space. Einstein showed that this
modification is equivalent to assuming that
photons are rigorously identical and that
it implied a "mysterious non-local interaction",
now understood as the requirement for a symmetric
quantum mechanical state. This work led to
the concept of coherent states and the development
of the laser. In the same papers, Einstein
extended Bose's formalism to material particles
(bosons) and predicted that they would condense
into their lowest quantum state at low enough
temperatures; this Bose–Einstein condensation
was observed experimentally in 1995. It was
later used by Lene Hau to slow, and then completely
stop, light in 1999 and 2001.The modern view
on this is that photons are, by virtue of
their integer spin, bosons (as opposed to
fermions with half-integer spin). By the spin-statistics
theorem, all bosons obey Bose–Einstein statistics
(whereas all fermions obey Fermi–Dirac statistics).
== Stimulated and spontaneous emission ==
In 1916, Albert Einstein showed that Planck's
radiation law could be derived from a semi-classical,
statistical treatment of photons and atoms,
which implies a link between the rates at
which atoms emit and absorb photons. The condition
follows from the assumption that functions
of the emission and absorption of radiation
by the atoms are independent of each other,
and that thermal equilibrium is made by way
of the radiation's interaction with the atoms.
Consider a cavity in thermal equilibrium with
all parts of itself and filled with electromagnetic
radiation and that the atoms can emit and
absorb that radiation. Thermal equilibrium
requires that the energy density
ρ
(
ν
)
{\displaystyle \rho (\nu )}
of photons with frequency
ν
{\displaystyle \nu }
(which is proportional to their number density)
is, on average, constant in time; hence, the
rate at which photons of any particular frequency
are emitted must equal the rate at which they
absorb them.Einstein began by postulating
simple proportionality relations for the different
reaction rates involved. In his model, the
rate
R
j
i
{\displaystyle R_{ji}}
for a system to absorb a photon of frequency
ν
{\displaystyle \nu }
and transition from a lower energy
E
j
{\displaystyle E_{j}}
to a higher energy
E
i
{\displaystyle E_{i}}
is proportional to the number
N
j
{\displaystyle N_{j}}
of atoms with energy
E
j
{\displaystyle E_{j}}
and to the energy density
ρ
(
ν
)
{\displaystyle \rho (\nu )}
of ambient photons of that frequency,
R
j
i
=
N
j
B
j
i
ρ
(
ν
)
{\displaystyle R_{ji}=N_{j}B_{ji}\rho (\nu
)\!}
where
B
j
i
{\displaystyle B_{ji}}
is the rate constant for absorption. For the
reverse process, there are two possibilities:
spontaneous emission of a photon, or the emission
of a photon initiated by the interaction of
the atom with a passing photon and the return
of the atom to the lower-energy state. Following
Einstein's approach, the corresponding rate
R
i
j
{\displaystyle R_{ij}}
for the emission of photons of frequency
ν
{\displaystyle \nu }
and transition from a higher energy
E
i
{\displaystyle E_{i}}
to a lower energy
E
j
{\displaystyle E_{j}}
is
R
i
j
=
N
i
A
i
j
+
N
i
B
i
j
ρ
(
ν
)
{\displaystyle R_{ij}=N_{i}A_{ij}+N_{i}B_{ij}\rho
(\nu )\!}
where
A
i
j
{\displaystyle A_{ij}}
is the rate constant for emitting a photon
spontaneously, and
B
i
j
{\displaystyle B_{ij}}
is the rate constant for emissions in response
to ambient photons (induced or stimulated
emission). In thermodynamic equilibrium, the
number of atoms in state i and those in state
j must, on average, be constant; hence, the
rates
R
j
i
{\displaystyle R_{ji}}
and
R
i
j
{\displaystyle R_{ij}}
must be equal. Also, by arguments analogous
to the derivation of Boltzmann statistics,
the ratio of
N
i
{\displaystyle N_{i}}
and
N
j
{\displaystyle N_{j}}
is
g
i
/
g
j
exp
⁡
(
E
j
−
E
i
)
/
(
k
T
)
,
{\displaystyle g_{i}/g_{j}\exp {(E_{j}-E_{i})/(kT)},}
where
g
i
,
j
{\displaystyle g_{i,j}}
are the degeneracy of the state i and that
of j, respectively,
E
i
,
j
{\displaystyle E_{i,j}}
their energies, k the Boltzmann constant and
T the system's temperature. From this, it
is readily derived that
g
i
B
i
j
=
g
j
B
j
i
{\displaystyle g_{i}B_{ij}=g_{j}B_{ji}}
and
A
i
j
=
8
π
h
ν
3
c
3
B
i
j
.
{\displaystyle A_{ij}={\frac {8\pi h\nu ^{3}}{c^{3}}}B_{ij}.}
The A and Bs are collectively known as the
Einstein coefficients.Einstein could not fully
justify his rate equations, but claimed that
it should be possible to calculate the coefficients
A
i
j
{\displaystyle A_{ij}}
,
B
j
i
{\displaystyle B_{ji}}
and
B
i
j
{\displaystyle B_{ij}}
once physicists had obtained "mechanics and
electrodynamics modified to accommodate the
quantum hypothesis". In fact, in 1926, Paul
Dirac derived the
B
i
j
{\displaystyle B_{ij}}
rate constants by using a semiclassical approach,
and, in 1927, succeeded in deriving all the
rate constants from first principles within
the framework of quantum theory. Dirac's work
was the foundation of quantum electrodynamics,
i.e., the quantization of the electromagnetic
field itself. Dirac's approach is also called
second quantization or quantum field theory;
earlier quantum mechanical treatments only
treat material particles as quantum mechanical,
not the electromagnetic field.
Einstein was troubled by the fact that his
theory seemed incomplete, since it did not
determine the direction of a spontaneously
emitted photon. A probabilistic nature of
light-particle motion was first considered
by Newton in his treatment of birefringence
and, more generally, of the splitting of light
beams at interfaces into a transmitted beam
and a reflected beam. Newton hypothesized
that hidden variables in the light particle
determined which of the two paths a single
photon would take. Similarly, Einstein hoped
for a more complete theory that would leave
nothing to chance, beginning his separation
from quantum mechanics. Ironically, Max Born's
probabilistic interpretation of the wave function
was inspired by Einstein's later work searching
for a more complete theory.
== Second quantization and high energy photon
interactions ==
In 1910, Peter Debye derived Planck's law
of black-body radiation from a relatively
simple assumption. He correctly decomposed
the electromagnetic field in a cavity into
its Fourier modes, and assumed that the energy
in any mode was an integer multiple of
h
ν
{\displaystyle h\nu }
, where
ν
{\displaystyle \nu }
is the frequency of the electromagnetic mode.
Planck's law of black-body radiation follows
immediately as a geometric sum. However, Debye's
approach failed to give the correct formula
for the energy fluctuations of blackbody radiation,
which were derived by Einstein in 1909.In
1925, Born, Heisenberg and Jordan reinterpreted
Debye's concept in a key way. As may be shown
classically, the Fourier modes of the electromagnetic
field—a complete set of electromagnetic
plane waves indexed by their wave vector k
and polarization state—are equivalent to
a set of uncoupled simple harmonic oscillators.
Treated quantum mechanically, the energy levels
of such oscillators are known to 
be
E
=
n
h
ν
{\displaystyle E=nh\nu }
, where
ν
{\displaystyle \nu }
is the oscillator frequency. The key new step
was to identify an electromagnetic mode with
energy
E
=
n
h
ν
{\displaystyle E=nh\nu }
as a state with
n
{\displaystyle n}
photons, each of energy
h
ν
{\displaystyle h\nu }
. This approach gives the correct energy fluctuation
formula.
Dirac took this one step further. He treated
the interaction between a charge and an electromagnetic
field as a small perturbation that induces
transitions in the photon states, changing
the numbers of photons in the modes, while
conserving energy and momentum overall. Dirac
was able to derive Einstein's
A
i
j
{\displaystyle A_{ij}}
and
B
i
j
{\displaystyle B_{ij}}
coefficients from first principles, and showed
that the Bose–Einstein statistics of photons
is a natural consequence of quantizing the
electromagnetic field correctly (Bose's reasoning
went in the opposite direction; he derived
Planck's law of black-body radiation by assuming
B–E statistics). In Dirac's time, it was
not yet known that all bosons, including photons,
must obey Bose–Einstein statistics.
Dirac's second-order perturbation theory can
involve virtual photons, transient intermediate
states of the electromagnetic field; the static
electric and magnetic interactions are mediated
by such virtual photons. In such quantum field
theories, the probability amplitude of observable
events is calculated by summing over all possible
intermediate steps, even ones that are unphysical;
hence, virtual photons are not constrained
to satisfy
E
=
p
c
{\displaystyle E=pc}
, and may have extra polarization states;
depending on the gauge used, virtual photons
may have three or four polarization states,
instead of the two states of real photons.
Although these transient virtual photons can
never be observed, they contribute measurably
to the probabilities of observable events.
Indeed, such second-order and higher-order
perturbation calculations can give apparently
infinite contributions to the sum. Such unphysical
results are corrected for using the technique
of renormalization.
Other virtual particles may contribute to
the summation as well; for example, two photons
may interact indirectly through virtual electron–positron
pairs. In fact, such photon–photon scattering
(see two-photon physics), as well as electron–photon
scattering, is meant to be one of the modes
of operations of the planned particle accelerator,
the International Linear Collider.In modern
physics notation, the quantum state of the
electromagnetic field is written as a Fock
state, a tensor product of the states for
each electromagnetic mode
|
n
k
0
⟩
⊗
|
n
k
1
⟩
⊗
⋯
⊗
|
n
k
n
⟩
…
{\displaystyle |n_{k_{0}}\rangle \otimes |n_{k_{1}}\rangle
\otimes \dots \otimes |n_{k_{n}}\rangle \dots
}
where
|
n
k
i
⟩
{\displaystyle |n_{k_{i}}\rangle }
represents the state in which
n
k
i
{\displaystyle \,n_{k_{i}}}
photons are in the mode
k
i
{\displaystyle k_{i}}
. In this notation, the creation of a new
photon in mode
k
i
{\displaystyle k_{i}}
(e.g., emitted from an atomic transition)
is written as
|
n
k
i
⟩
→
|
n
k
i
+
1
⟩
{\displaystyle |n_{k_{i}}\rangle \rightarrow
|n_{k_{i}}+1\rangle }
. This notation merely expresses the concept
of Born, Heisenberg and Jordan described above,
and does not add any physics.
== The hadronic properties of the photon ==
Measurements of the interaction between energetic
photons and hadrons show that the interaction
is much more intense than expected by the
interaction of merely photons with the hadron's
electric charge. Furthermore, the interaction
of energetic photons with protons is similar
to the interaction of photons with neutrons
in spite of the fact that the electric charge
structures of protons and neutrons are substantially
different. A theory called Vector Meson Dominance
(VMD) was developed to explain this effect.
According to VMD, the photon is a superposition
of the pure electromagnetic photon which interacts
only with electric charges and vector mesons.
However, if experimentally probed at very
short distances, the intrinsic structure of
the photon is recognized as a flux of quark
and gluon components, quasi-free according
to asymptotic freedom in QCD and described
by the photon structure function. A comprehensive
comparison of data with theoretical predictions
was presented in a review in 2000.
== The photon as a gauge boson ==
The electromagnetic field can be understood
as a gauge field, i.e., as a field that results
from requiring that a gauge symmetry holds
independently at every position in spacetime.
For the electromagnetic field, this gauge
symmetry is the Abelian U(1) symmetry of complex
numbers of absolute value 1, which reflects
the ability to vary the phase of a complex
field without affecting observables or real
valued functions made from it, such as the
energy or the Lagrangian.
The quanta of an Abelian gauge field must
be massless, uncharged bosons, as long as
the symmetry is not broken; hence, the photon
is predicted to be massless, and to have zero
electric charge and integer spin. The particular
form of the electromagnetic interaction specifies
that the photon must have spin ±1; thus,
its helicity must be
±
ℏ
{\displaystyle \pm \hbar }
. These two spin components correspond to
the classical concepts of right-handed and
left-handed circularly polarized light. However,
the transient virtual photons of quantum electrodynamics
may also adopt unphysical polarization states.In
the prevailing Standard Model of physics,
the photon is one of four gauge bosons in
the electroweak interaction; the other three
are denoted W+, W− and Z0 and are responsible
for the weak interaction. Unlike the photon,
these gauge bosons have mass, owing to a mechanism
that breaks their SU(2) gauge symmetry. The
unification of the photon with W and Z gauge
bosons in the electroweak interaction was
accomplished by Sheldon Glashow, Abdus Salam
and Steven Weinberg, for which they were awarded
the 1979 Nobel Prize in physics. Physicists
continue to hypothesize grand unified theories
that connect these four gauge bosons with
the eight gluon gauge bosons of quantum chromodynamics;
however, key predictions of these theories,
such as proton decay, have not been observed
experimentally.
== Contributions to the mass of a system ==
The energy of a system that emits a photon
is decreased by the energy
E
{\displaystyle E}
of the photon as measured in the rest frame
of the emitting system, which may result in
a reduction in mass in the amount
E
/
c
2
{\displaystyle {E}/{c^{2}}}
. Similarly, the mass of a system that absorbs
a photon is increased by a corresponding amount.
As an application, the energy balance of nuclear
reactions involving photons is commonly written
in terms of the masses of the nuclei involved,
and terms of the form
E
/
c
2
{\displaystyle {E}/{c^{2}}}
for the gamma photons (and for other relevant
energies, such as the recoil energy of nuclei).This
concept is applied in key predictions of quantum
electrodynamics (QED, see above). In that
theory, the mass of electrons (or, more generally,
leptons) is modified by including the mass
contributions of virtual photons, in a technique
known as renormalization. Such "radiative
corrections" contribute to a number of predictions
of QED, such as the magnetic dipole moment
of leptons, the Lamb shift, and the hyperfine
structure of bound lepton pairs, such as muonium
and positronium.Since photons contribute to
the stress–energy tensor, they exert a gravitational
attraction on other objects, according to
the theory of general relativity. Conversely,
photons are themselves affected by gravity;
their normally straight trajectories may be
bent by warped spacetime, as in gravitational
lensing, and their frequencies may be lowered
by moving to a higher gravitational potential,
as in the Pound–Rebka experiment. However,
these effects are not specific to photons;
exactly the same effects would be predicted
for classical electromagnetic waves.
== Photons in matter ==
Light that travels through transparent matter
does so at a lower speed than c, the speed
of light in a vacuum. For example, photons
engage in so many collisions on the way from
the core of the sun that radiant energy can
take about a million years to reach the surface;
however, once in open space, a photon takes
only 8.3 minutes to reach Earth. The factor
by which the speed is decreased is called
the refractive index of the material. In a
classical wave picture, the slowing can be
explained by the light inducing electric polarization
in the matter, the polarized matter radiating
new light, and that new light interfering
with the original light wave to form a delayed
wave. In a particle picture, the slowing can
instead be described as a blending of the
photon with quantum excitations of the matter
to produce quasi-particles known as polariton
(other quasi-particles are phonons and excitons);
this polariton has a nonzero effective mass,
which means that it cannot travel at c. Light
of different frequencies may travel through
matter at different speeds; this is called
dispersion (not to be confused with scattering).
In some cases, it can result in extremely
slow speeds of light in matter. The effects
of photon interactions with other quasi-particles
may be observed directly in Raman scattering
and Brillouin scattering.Photons can also
be absorbed by nuclei, atoms or molecules,
provoking transitions between their energy
levels. A classic example is the molecular
transition of retinal (C20H28O), which is
responsible for vision, as discovered in 1958
by Nobel laureate biochemist George Wald and
co-workers. The absorption provokes a cis-trans
isomerization that, in combination with other
such transitions, is transduced into nerve
impulses. The absorption of photons can even
break chemical bonds, as in the photodissociation
of chlorine; this is the subject of photochemistry.
== Technological applications ==
Photons have many applications in technology.
These examples are chosen to illustrate applications
of photons per se, rather than general optical
devices such as lenses, etc. that could operate
under a classical theory of light. The laser
is an extremely important application and
is discussed above under stimulated emission.
Individual photons can be detected by several
methods. The classic photomultiplier tube
exploits the photoelectric effect: a photon
of sufficient energy strikes a metal plate
and knocks free an electron, initiating an
ever-amplifying avalanche of electrons. Semiconductor
charge-coupled device chips use a similar
effect: an incident photon generates a charge
on a microscopic capacitor that can be detected.
Other detectors such as Geiger counters use
the ability of photons to ionize gas molecules
contained in the device, causing a detectable
change of conductivity of the gas.Planck's
energy formula
E
=
h
ν
{\displaystyle E=h\nu }
is often used by engineers and chemists in
design, both to compute the change in energy
resulting from a photon absorption and to
determine the frequency of the light emitted
from a given photon emission. For example,
the emission spectrum of a gas-discharge lamp
can be altered by filling it with (mixtures
of) gases with different electronic energy
level configurations.
Under some conditions, an energy transition
can be excited by "two" photons that individually
would be insufficient. This allows for higher
resolution microscopy, because the sample
absorbs energy only in the spectrum where
two beams of different colors overlap significantly,
which can be made much smaller than the excitation
volume of a single beam (see two-photon excitation
microscopy). Moreover, these photons cause
less damage to the sample, since they are
of lower energy.In some cases, two energy
transitions can be coupled so that, as one
system absorbs a photon, another nearby system
"steals" its energy and re-emits a photon
of a different frequency. This is the basis
of fluorescence resonance energy transfer,
a technique that is used in molecular biology
to study the interaction of suitable proteins.Several
different kinds of hardware random number
generators involve the detection of single
photons. In one example, for each bit in the
random sequence that is to be produced, a
photon is sent to a beam-splitter. In such
a situation, there are two possible outcomes
of equal probability. The actual outcome is
used to determine whether the next bit in
the sequence is "0" or "1".
== Recent research ==
Much research has been devoted to applications
of photons in the field of quantum optics.
Photons seem well-suited to be elements of
an extremely fast quantum computer, and the
quantum entanglement of photons is a focus
of research. Nonlinear optical processes are
another active research area, with topics
such as two-photon absorption, self-phase
modulation, modulational instability and optical
parametric oscillators. However, such processes
generally do not require the assumption of
photons per se; they may often be modeled
by treating atoms as nonlinear oscillators.
The nonlinear process of spontaneous parametric
down conversion is often used to produce single-photon
states. Finally, photons are essential in
some aspects of optical communication, especially
for quantum cryptography.Two-photon physics
studies interactions between photons, which
are rare. In 2018, MIT researchers announced
the discovery of bound photon triplets, which
may involve polaritons.
== See also ==
== 
Notes
