In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, specifically in
quantum mechanics
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, ...
, a coherent state is the specific
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
of the
, often described as a state which has dynamics most closely resembling the oscillatory behavior of a
classical harmonic oscillator. It was the first example of
quantum dynamics In physics, quantum dynamics is the quantum version of classical dynamics. Quantum dynamics deals with the motions, and energy and momentum exchanges of systems whose behavior is governed by the laws of quantum mechanics. Quantum dynamics is relevan ...
when
Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theory ...
derived it in 1926, while searching for solutions 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 ...
that satisfy the
correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says t ...
.
The quantum harmonic oscillator (and hence the coherent states) arise in the quantum theory of a wide range of physical systems.
[J.R. Klauder and B. Skagerstam, ''Coherent States'', World Scientific, Singapore, 1985.] For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic
potential well
A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy (kinetic energy in the case of a gravitational potential well) because it is captur ...
(for an early reference, see e.g.
Schiff's textbook). The coherent state describes a state in a system for which the ground-state wavepacket is displaced from the origin of the system. This state can be related to classical solutions by a particle oscillating with an amplitude equivalent to the displacement.
These states, expressed as ''
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of the
lowering operator
In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an Operator (mathematics), operator that increases or decreases the eigenvalue of another operator. In quantu ...
'' and forming an ''
overcomplete'' family, were introduced in the early papers of
John R. Klauder, e. g.
In the quantum theory of light (
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 other
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
ic
quantum field theories
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 ...
, coherent states were introduced by the work of
Roy J. Glauber
Roy Jay Glauber (September 1, 1925 – December 26, 2018) was an American theoretical physicist. He was the Mallinckrodt Professor of Physics at Harvard University and Adjunct Professor of Optical Sciences at the University of Arizona. Born in New ...
in 1963 and are also known as Glauber states.
The concept of coherent states has been considerably abstracted; it has become a major topic in
mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
and in
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
, with applications ranging from
quantization to
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
and
image processing
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
(see
Coherent states in mathematical physics). For this reason, the coherent states associated to the
are sometimes referred to as ''canonical coherent states'' (CCS), ''standard coherent states'', ''Gaussian'' states, or oscillator states.
Coherent states in quantum optics
In
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 ...
the coherent state refers to a state of the quantized
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
, etc.
[ J-P. Gazeau, ''Coherent States in Quantum Physics'', Wiley-VCH, Berlin, 2009.] that describes a maximal kind of
coherence and a classical kind of behavior.
Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theory ...
derived it as a "minimum
uncertainty
Uncertainty refers to epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially observable or ...
"
Gaussian wavepacket in 1926, searching for solutions 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 ...
that satisfy the
correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says t ...
.
It is a minimum uncertainty state, with the single free parameter chosen to make the relative dispersion (standard deviation in natural dimensionless units) equal for position and momentum, each being equally small at high energy.
Further, in contrast to the
energy eigenstates
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 ...
of the system, the time evolution of a coherent state is concentrated along the classical
trajectories
A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete traj ...
. The quantum linear harmonic oscillator, and hence coherent states, arise in the quantum theory of a wide range of physical systems. They occur in the quantum theory of light (
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 other
bosonic
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
quantum field theories
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 ...
.
While minimum uncertainty Gaussian wave-packets had been well-known, they did not attract full attention until
Roy J. Glauber
Roy Jay Glauber (September 1, 1925 – December 26, 2018) was an American theoretical physicist. He was the Mallinckrodt Professor of Physics at Harvard University and Adjunct Professor of Optical Sciences at the University of Arizona. Born in New ...
, in 1963, provided a complete quantum-theoretic description of coherence in the electromagnetic field. In this respect, the concurrent contribution of
E.C.G. Sudarshan should not be omitted, (there is, however, a note in Glauber's paper that reads: "Uses of these states as
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
s for the
-quantum states have, however, been made by J. Schwinger).
Glauber was prompted to do this to provide a description of the
Hanbury-Brown & Twiss experiment which generated very wide baseline (hundreds or thousands of miles)
interference patterns that could be used to determine stellar diameters. This opened the door to a much more comprehensive understanding of coherence. (For more, see
Quantum mechanical description.)
In classical
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
, light is thought of as
electromagnetic waves
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) lig ...
radiating from a source. Often, coherent laser light is thought of as light that is emitted by many such sources that are in
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 ...
. Actually, the picture of one
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 ...
being in-phase with another is not valid in quantum theory. Laser radiation is produced in a
resonant cavity
A resonator is a device or system that exhibits resonance or resonant behavior. That is, it naturally oscillates with greater amplitude at some frequencies, called resonant frequencies, than at other frequencies. The oscillations in a resonato ...
where the
resonant frequency
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillatin ...
of the cavity is the same as the frequency associated with the
atomic electron transition
Atomic electron transition is a change (or jump) of an electron from one energy level to another within an atom or artificial atom. It appears discontinuous as the electron "jumps" from one quantized energy level to another, typically in a few na ...
s providing energy flow into the field. As energy in the resonant mode builds up, the probability 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 ...
, in that mode only, increases. That is a positive
feedback loop
Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled c ...
in which the amplitude in the resonant mode
increases exponentially until some
non-linear effects limit it. As a counter-example, a
light bulb
An electric light, lamp, or light bulb is an electrical component that produces light. It is the most common form of artificial lighting. Lamps usually have a base made of ceramic, metal, glass, or plastic, which secures the lamp in the soc ...
radiates light into a continuum of modes, and there is nothing that selects any one mode over the other. The emission process is highly random in space and time (see
thermal light
Thermal radiation is electromagnetic radiation generated by the thermal motion of particles in matter. Thermal radiation is generated when heat from the movement of charges in the material (electrons and protons in common forms of matter) is ...
). In a
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 ...
, however, light is emitted into a resonant mode, and that mode is highly
coherent
Coherence, coherency, or coherent may refer to the following:
Physics
* Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference
* Coherence (units of measurement), a deri ...
. Thus, laser light is idealized as a coherent state. (Classically we describe such a state by an
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
oscillating as a stable wave. See Fig.1)
Besides describing lasers, coherent states also behave in a convenient manner when describing the quantum action of
beam splitter
A beam splitter or ''beamsplitter'' is an optical device that splits a beam of light into a transmitted and a reflected beam. It is a crucial part of many optical experimental and measurement systems, such as interferometers, also finding wide ...
s: two coherent-state input beams will simply convert to two coherent-state beams at the output with new amplitudes given by classical electromagnetic wave formulas;
such a simple behaviour does not occur for other input states, including number states. Likewise if a coherent-state light beam is partially absorbed, then the remainder is a pure coherent state with a smaller amplitude, whereas partial absorption of non-coherent-state light produces a more complicated statistical
mixed state.
[ Thermal light can be described as a statistical mixture of coherent states, and the typical way of defining ]nonclassical light
Nonclassical light is light that cannot be described using classical electromagnetism; its characteristics are described by the quantized electromagnetic field and quantum mechanics.
The most common described forms of nonclassical light are the fo ...
is that it cannot be described as a simple statistical mixture of coherent states.[
The energy eigenstates of the linear harmonic oscillator (e.g., masses on springs, lattice vibrations in a solid, vibrational motions of nuclei in molecules, or oscillations in the electromagnetic field) are fixed-number quantum states. The ]Fock state
In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles (or quanta). These states are named after the Soviet physicist Vladimir Fock. Fock states play an impo ...
(e.g. a single photon) is the most particle-like state; it has a fixed number of particles, and phase is indeterminate. A coherent state distributes its quantum-mechanical uncertainty equally between the canonically conjugate coordinates, position and momentum, and the relative uncertainty in phase efined_heuristically.html"_;"title="heuristic.html"_;"title="efined_heuristic">efined_heuristically">heuristic.html"_;"title="efined_heuristic">efined_heuristicallyand_amplitude_are_roughly_equal—and_small_at_high_amplitude.
_Quantum_mechanical_definition
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