Coherence (physics)

Encyclopedia

In physics

,

s that enables stationary (i.e. temporally and spatially constant) interference. More generally, coherence describes all properties of the correlation

between physical quantities of a wave.

When interfering, two waves can add together to create a wave of greater amplitude than either one (

. Two waves are said to be coherent if they have a constant relative phase. The degree of coherence is measured by the interference visibility, a measure of how perfectly the waves can cancel due to destructive interference. Cancellation is virtual or local since a wave cannot have negative energy.

's double-slit experiment

in optics

but is now used in any field that involves waves, such as acoustics

, electrical engineering

, neuroscience

, and quantum mechanics

. The property of coherence is the basis for commercial applications such as holography

, the Sagnac gyroscope

, radio

antenna array

s, optical coherence tomography

and telescope interferometers (astronomical optical interferometers and radio telescope

s).

function. The cross-correlation quantifies the ability to predict the value of the second wave by knowing the value of the first. As an example, consider two waves perfectly correlated for all times. At any time, if the first wave changes, the second will change in the same way. If combined they can exhibit complete constructive interference/superposition at all times, then it follows that they are perfectly coherent. As will be discussed below, the second wave need not be a separate entity. It could be the first wave at a different time or position. In this case, the measure of correlation is the autocorrelation

function (sometimes called

or some generalization thereof.

In most of these systems, one can measure the wave directly. Consequently, its correlation with another wave can simply be calculated. However, in optics one cannot measure the electric field

directly as it oscillates much faster than any detector’s time resolution. Instead, we measure the intensity

of the light. Most of the concepts involving coherence which will be introduced below were developed in the field of optics and then used in other fields. Therefore, many of the standard measurements of coherence are indirect measurements, even in fields where the wave can be measured directly.

One should be careful not to confuse the coherence time with the time duration of the signal, nor the coherence length with the coherence area (see below).

.

Formally, this follows from the convolution theorem

in mathematics, which relates the Fourier transform

of the power spectrum (the intensity of each frequency) to its autocorrelation

.

The most monochromatic sources are usually laser

s; such high monochromaticity implies long coherence lengths (up to hundreds of meters). For example, a stabilized helium-neon laser

can produce light with coherence lengths in excess of 5 m. Not all lasers are monochromatic, however (e.g. for a mode-locked Ti-sapphire laser

, Δλ ≈ 2 nm - 70 nm). LEDs are characterized by Δλ ≈ 50 nm, and tungsten filament lights exhibit Δλ ≈ 600 nm, so these sources have shorter coherence times than the most monochromatic lasers.

Holography

requires light with a long coherence time. In contrast, Optical coherence tomography

uses light with a short coherence time.

or Mach–Zehnder interferometer. In these devices, a wave is combined with a copy of itself that is delayed by time τ. A detector measures the time-averaged intensity

of the light exiting the interferometer. The resulting interference visibility (e.g. see Figure 4) gives the temporal coherence at delay τ. Since for most natural light sources, the coherence time is much shorter than the time resolution of any detector, the detector itself does the time averaging. Consider the example shown in Figure 3. At a fixed delay, here 2τ

between two points in a wave for all times. If a wave has only 1 value of amplitude over an infinite length, it is perfectly spatially coherent. The range of separation between the two points over which there is significant interference is called the coherence area,

, has large spatial coherence because antennas at opposite ends of the array emit with a fixed phase-relationship. Light waves produced by a laser often have high temporal and spatial coherence (though the degree of coherence depends strongly on the exact properties of the laser). Spatial coherence of laser beams also manifests itself as speckle patterns and diffraction fringes seen at the edges of shadow.

Holography requires temporally and spatially coherent light. Its inventor, Dennis Gabor

, produced successful holograms more than ten years before lasers were invented. To produce coherent light he passed the monochromatic light from an emission line of a mercury-vapor lamp

through a pinhole spatial filter.

In February 2011, Dr Andrew Truscott, leader of a research team at the ARC Centre of Excellence for Quantum-Atom Optics at Australian National University

in Canberra, Australian Capital Territory, showed that helium

atoms cooled to near absolute zero

/ Bose-Einstein condensate state, can be made to flow and behave as a coherent beam as occurs in a laser.

). Conversely, if waves of different frequencies are not coherent, then, when combined, they create a wave that is continuous in time (e.g. white light or white noise

). The temporal duration of the pulse is limited by the spectral bandwidth of the light according to:

,

which follows from the properties of the Fourier transform (for quantum particles it also results in the Heisenberg uncertainty principle).

If the phase depends linearly on the frequency (i.e. ) then the pulse will have the minimum time duration for its bandwidth (a

).

optical interferometer, such as an intensity optical correlator

, frequency-resolved optical gating

(FROG), or Spectral phase interferometry for direct electric-field reconstruction

(SPIDER).

rotated to any angle will always transmit half the incident intensity when averaged over time.

If the electric field wanders by a smaller amount the light will be partially polarized so that at some angle, the polarizer will transmit more than half the intensity. If a wave is combined with an orthogonally polarized copy of itself delayed by less than the coherence time, partially polarized light is created.

The polarization of a light beam is represented by a vector in the Poincare sphere. For polarized light the end of the vector lies on the surface of the sphere, whereas the vector has zero length for unpolarized light. The vector for partially polarized light lies within the sphere

. Holographic objects are used frequently in daily life in bank notes and credit cards.

. Additionally the problems of the following subchapter are treated.

, all objects have wave-like properties (see de Broglie waves). For instance, in Young's double-slit experiment electrons can be used in the place of light waves. Each electron can go through either slit and hence has two paths that it can take to a particular final position. In quantum mechanics these two paths interfere. If there is destructive interference, the electron never arrives at that particular position. This ability to interfere is indicative of quantum coherence.

The quantum description of perfectly coherent paths is called a pure state, in which the two paths are combined in a superposition

. The correlation between the two particles exceeds what would be predicted for classical correlation alone (see Bell's inequalities). If this two-particle system is decohered (which would occur in a measurement via Einselection

), then there is no longer any phase relationship between the two states. The quantum description of imperfectly coherent paths is called a mixed state, described by a density matrix

(also called the "statistical operator") and analogous to a classical system of mixed probabilities. It has long been considered that, for mixed states, all correlations were entirely classical; however, in more recent work since 2001 it has been found that quantum correlations are present in certain mixed separable states and that such nonclassical correlations can be described within the conceptual framework of the so-called quantum discord

.

Large-scale (macroscopic

) quantum coherence leads to novel phenomena. For instance, the laser

, superconductivity

, and superfluidity are examples of highly coherent quantum systems, whose effects are evident at the macroscopic scale. These examples of quantum coherence are Bose–Einstein condensate

s. Here, all the particles that make up the condensate are in-phase; they are thus necessarily all described by a single quantum wavefunction.

On the other hand, the Schrödinger's cat

thought experiment, highlights the fact that quantum coherence is not typically seen at the macroscopic scale but has been observed in the motion of a mechanical resonator (see Quantum machine

).

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

,

**coherence**is a property of waveWave

In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...

s that enables stationary (i.e. temporally and spatially constant) interference. More generally, coherence describes all properties of the correlation

Correlation function

A correlation function is the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points...

between physical quantities of a wave.

When interfering, two waves can add together to create a wave of greater amplitude than either one (

**constructive interference**) or subtract from each other to create a wave of lesser amplitude than either one (**destructive interference**), depending on their relative phasePhase (waves)

Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

. Two waves are said to be coherent if they have a constant relative phase. The degree of coherence is measured by the interference visibility, a measure of how perfectly the waves can cancel due to destructive interference. Cancellation is virtual or local since a wave cannot have negative energy.

## Introduction

Coherence was originally introduced in connection with Thomas YoungThomas Young (scientist)

Thomas Young was an English polymath. He is famous for having partly deciphered Egyptian hieroglyphics before Jean-François Champollion eventually expanded on his work...

's double-slit experiment

Double-slit experiment

The double-slit experiment, sometimes called Young's experiment, is a demonstration that matter and energy can display characteristics of both waves and particles...

in optics

Optics

Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

but is now used in any field that involves waves, such as acoustics

Acoustics

Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...

, electrical engineering

Electrical engineering

Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...

, neuroscience

Neuroscience

Neuroscience is the scientific study of the nervous system. Traditionally, neuroscience has been seen as a branch of biology. However, it is currently an interdisciplinary science that collaborates with other fields such as chemistry, computer science, engineering, linguistics, mathematics,...

, and quantum mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

. The property of coherence is the basis for commercial applications such as holography

Holography

Holography is a technique that allows the light scattered from an object to be recorded and later reconstructed so that when an imaging system is placed in the reconstructed beam, an image of the object will be seen even when the object is no longer present...

, the Sagnac gyroscope

Gyroscope

A gyroscope is a device for measuring or maintaining orientation, based on the principles of angular momentum. In essence, a mechanical gyroscope is a spinning wheel or disk whose axle is free to take any orientation...

, radio

Radio

Radio is the transmission of signals through free space by modulation of electromagnetic waves with frequencies below those of visible light. Electromagnetic radiation travels by means of oscillating electromagnetic fields that pass through the air and the vacuum of space...

antenna array

Phased array

In wave theory, a phased array is an array of antennas in which the relative phases of the respective signals feeding the antennas are varied in such a way that the effective radiation pattern of the array is reinforced in a desired direction and suppressed in undesired directions.An antenna array...

s, optical coherence tomography

Optical coherence tomography

Optical coherence tomography is an optical signal acquisition and processing method. It captures micrometer-resolution, three-dimensional images from within optical scattering media . Optical coherence tomography is an interferometric technique, typically employing near-infrared light...

and telescope interferometers (astronomical optical interferometers and radio telescope

Radio telescope

A radio telescope is a form of directional radio antenna used in radio astronomy. The same types of antennas are also used in tracking and collecting data from satellites and space probes...

s).

## Coherence and correlation

The coherence of two waves follows from how well correlated the waves are as quantified by the cross-correlationCross-correlation

In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long-duration signal for a shorter, known feature...

function. The cross-correlation quantifies the ability to predict the value of the second wave by knowing the value of the first. As an example, consider two waves perfectly correlated for all times. At any time, if the first wave changes, the second will change in the same way. If combined they can exhibit complete constructive interference/superposition at all times, then it follows that they are perfectly coherent. As will be discussed below, the second wave need not be a separate entity. It could be the first wave at a different time or position. In this case, the measure of correlation is the autocorrelation

Autocorrelation

Autocorrelation is the cross-correlation of a signal with itself. Informally, it is the similarity between observations as a function of the time separation between them...

function (sometimes called

**self-coherence**). Degree of correlation involves correlation functions.## Examples of wave-like states

These states are unified by the fact that their behavior is described by a wave equationWave equation

The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...

or some generalization thereof.

- Waves in a rope (up and down) or slinkySlinkySlinky or "Lazy Spring" is a toy consisting of a helical spring that stretches and can bounce up and down. It can perform a number of tricks, including traveling down a flight of steps end-over-end as it stretches and re-forms itself with the aid of gravity and its own momentum.-History:The toy was...

(compression and expansion) - Surface waves in a liquid
- Electric signals (fields) in transmission cables
- SoundSoundSound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.-Propagation of...
- Radio waveRadio WaveRadio Wave may refer to:*Radio frequency*Radio Wave 96.5, a radio station in Blackpool, UK...

s and Microwaves - Light waves (opticsOpticsOptics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

) - ElectronElectronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s, atomAtomThe atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

s and any other object (such as a baseball, as described by quantum physics)

In most of these systems, one can measure the wave directly. Consequently, its correlation with another wave can simply be calculated. However, in optics one cannot measure the electric field

Electric field

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

directly as it oscillates much faster than any detector’s time resolution. Instead, we measure the intensity

Intensity (physics)

In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...

of the light. Most of the concepts involving coherence which will be introduced below were developed in the field of optics and then used in other fields. Therefore, many of the standard measurements of coherence are indirect measurements, even in fields where the wave can be measured directly.

## Temporal coherence

Temporal coherence is the measure of the average correlation between the value of a wave and itself delayed by τ, at any pair of times. Temporal coherence tells us how monochromatic a source is. In other words, it characterizes how well a wave can interfere with itself at a different time. The delay over which the phase or amplitude wanders by a significant amount (and hence the correlation decreases by significant amount) is defined as the coherence timeCoherence time

For an electromagnetic wave, the coherence time is the time over which a propagating wave may be considered coherent...

*τ*. At τ=0 the degree of coherence is perfect whereas it drops significantly by delay_{c}*τ*. The coherence length_{c}Coherence length

In physics, coherence length is the propagation distance from a coherent source to a point where an electromagnetic wave maintains a specified degree of coherence. The significance is that interference will be strong within a coherence length of the source, but not beyond it...

*L*is defined as the distance the wave travels in time τ_{c}_{c}.One should be careful not to confuse the coherence time with the time duration of the signal, nor the coherence length with the coherence area (see below).

### The relationship between coherence time and bandwidth

It can be shown that the faster a wave decorrelates (and hence the smaller τ_{c}is) the larger the range of frequencies Δf the wave contains. Thus there is a tradeoff:.

Formally, this follows from the convolution theorem

Convolution theorem

In mathematics, the convolution theorem states that under suitableconditions the Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain equals point-wise multiplication in the other domain...

in mathematics, which relates the Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

of the power spectrum (the intensity of each frequency) to its autocorrelation

Autocorrelation

Autocorrelation is the cross-correlation of a signal with itself. Informally, it is the similarity between observations as a function of the time separation between them...

.

### Examples of temporal coherence

We consider four examples of temporal coherence.- A wave containing only a single frequency (monochromatic) is perfectly correlated at all times according to the above relation. (See Figure 1)
- Conversely, a wave whose phase drifts quickly will have a short coherence time. (See Figure 2)
- Similarly, pulses (wave packetWave packetIn physics, a wave packet is a short "burst" or "envelope" of wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere...

s) of waves, which naturally have a broad range of frequencies, also have a short coherence time since the amplitude of the wave changes quickly. (See Figure 3) - Finally, white light, which has a very broad range of frequencies, is a wave which varies quickly in both amplitude and phase. Since it consequently has a very short coherence time (just 10 periods or so), it is often called incoherent.

The most monochromatic sources are usually laser

Laser

A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

s; such high monochromaticity implies long coherence lengths (up to hundreds of meters). For example, a stabilized helium-neon laser

Helium-neon laser

A helium–neon laser or HeNe laser, is a type of gas laser whose gain medium consists of a mixture of helium and neon inside of a small bore capillary tube, usually excited by a DC electrical discharge.- History of HeNe laser development:...

can produce light with coherence lengths in excess of 5 m. Not all lasers are monochromatic, however (e.g. for a mode-locked Ti-sapphire laser

Ti-sapphire laser

Ti:sapphire lasers are tunable lasers which emit red and near-infrared light in the range from 650 to 1100 nanometers. These lasers are mainly used in scientific research because of their tunability and their ability to generate ultrashort pulses...

, Δλ ≈ 2 nm - 70 nm). LEDs are characterized by Δλ ≈ 50 nm, and tungsten filament lights exhibit Δλ ≈ 600 nm, so these sources have shorter coherence times than the most monochromatic lasers.

Holography

Holography

Holography is a technique that allows the light scattered from an object to be recorded and later reconstructed so that when an imaging system is placed in the reconstructed beam, an image of the object will be seen even when the object is no longer present...

requires light with a long coherence time. In contrast, Optical coherence tomography

Optical coherence tomography

Optical coherence tomography is an optical signal acquisition and processing method. It captures micrometer-resolution, three-dimensional images from within optical scattering media . Optical coherence tomography is an interferometric technique, typically employing near-infrared light...

uses light with a short coherence time.

### Measurement of temporal coherence

In optics, temporal coherence is measured in an interferometer such as the Michelson interferometerMichelson interferometer

The Michelson interferometer is the most common configuration for optical interferometry and was invented by Albert Abraham Michelson. An interference pattern is produced by splitting a beam of light into two paths, bouncing the beams back and recombining them...

or Mach–Zehnder interferometer. In these devices, a wave is combined with a copy of itself that is delayed by time τ. A detector measures the time-averaged intensity

Intensity (physics)

In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...

of the light exiting the interferometer. The resulting interference visibility (e.g. see Figure 4) gives the temporal coherence at delay τ. Since for most natural light sources, the coherence time is much shorter than the time resolution of any detector, the detector itself does the time averaging. Consider the example shown in Figure 3. At a fixed delay, here 2τ

_{c}, an infinitely fast detector would measure an intensity that fluctuates significantly over a time*t*equal to τ_{c}. In this case, to find the temporal coherence at 2τ_{c}, one would manually time-average the intensity.## Spatial coherence

In some systems, such as water waves or optics, wave-like states can extend over one or two dimensions. Spatial coherence describes the ability for two points in space,*x*and_{1}*x*, in the extent of a wave to interfere, when averaged over time. More precisely, the spatial coherence is the cross-correlation_{2}Cross-correlation

In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long-duration signal for a shorter, known feature...

between two points in a wave for all times. If a wave has only 1 value of amplitude over an infinite length, it is perfectly spatially coherent. The range of separation between the two points over which there is significant interference is called the coherence area,

*A*. This is the relevant type of coherence for the Young’s double-slit interferometer. It is also used in optical imaging systems and particularly in various types of astronomy telescopes. Sometimes people also use “spatial coherence” to refer to the visibility when a wave-like state is combined with a spatially shifted copy of itself._{c}### Examples of spatial coherence

Consider a tungsten light-bulb filament. Different points in the filament emit light independently and have no fixed phase-relationship. In detail, at any point in time the profile of the emitted light is going to be distorted. The profile will change randomly over the coherence time . Since for a white-light source such as a light-bulb is small, the filament is considered a spatially incoherent source. In contrast, a radio antenna arrayPhased array

In wave theory, a phased array is an array of antennas in which the relative phases of the respective signals feeding the antennas are varied in such a way that the effective radiation pattern of the array is reinforced in a desired direction and suppressed in undesired directions.An antenna array...

, has large spatial coherence because antennas at opposite ends of the array emit with a fixed phase-relationship. Light waves produced by a laser often have high temporal and spatial coherence (though the degree of coherence depends strongly on the exact properties of the laser). Spatial coherence of laser beams also manifests itself as speckle patterns and diffraction fringes seen at the edges of shadow.

Holography requires temporally and spatially coherent light. Its inventor, Dennis Gabor

Dennis Gabor

Dennis Gabor CBE, FRS was a Hungarian-British electrical engineer and inventor, most notable for inventing holography, for which he later received the 1971 Nobel Prize in Physics....

, produced successful holograms more than ten years before lasers were invented. To produce coherent light he passed the monochromatic light from an emission line of a mercury-vapor lamp

Mercury-vapor lamp

A mercury-vapor lamp is a gas discharge lamp that uses an electric arc through vaporized mercury to produce light. The arc discharge is generally confined to a small fused quartz arc tube mounted within a larger borosilicate glass bulb...

through a pinhole spatial filter.

In February 2011, Dr Andrew Truscott, leader of a research team at the ARC Centre of Excellence for Quantum-Atom Optics at Australian National University

Australian National University

The Australian National University is a teaching and research university located in the Australian capital, Canberra.As of 2009, the ANU employs 3,945 administrative staff who teach approximately 10,000 undergraduates, and 7,500 postgraduate students...

in Canberra, Australian Capital Territory, showed that helium

Helium

Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...

atoms cooled to near absolute zero

Absolute zero

Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

/ Bose-Einstein condensate state, can be made to flow and behave as a coherent beam as occurs in a laser.

## Spectral coherence

Waves of different frequencies (in light these are different colours) can interfere to form a pulse if they have a fixed relative phase-relationship (see Fourier transformFourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

). Conversely, if waves of different frequencies are not coherent, then, when combined, they create a wave that is continuous in time (e.g. white light or white noise

White noise

White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...

). The temporal duration of the pulse is limited by the spectral bandwidth of the light according to:

,

which follows from the properties of the Fourier transform (for quantum particles it also results in the Heisenberg uncertainty principle).

If the phase depends linearly on the frequency (i.e. ) then the pulse will have the minimum time duration for its bandwidth (a

*transform-limited*pulse), otherwise it is chirped (see dispersionDispersion (optics)

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency.Media having such a property are termed dispersive media...

).

### Measurement of spectral coherence

Measurement of the spectral coherence of light requires a nonlinearNonlinear optics

Nonlinear optics is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light...

optical interferometer, such as an intensity optical correlator

Optical autocorrelation

In optics, various autocorrelation functions can be experimentally realized. The field autocorrelation may be used to calculate the spectrum of a source of light, while the intensity autocorrelation and the interferometric autocorrelation are commonly used to estimate the duration of ultrashort...

, frequency-resolved optical gating

Frequency-resolved optical gating

In optics, frequency-resolved optical gating is a derivative of autocorrelation, but is far superior in its ability to measure ultrafast optical pulse shapes...

(FROG), or Spectral phase interferometry for direct electric-field reconstruction

Spectral phase interferometry for direct electric-field reconstruction

In ultrafast optics, spectral phase interferometry for direct electric-field reconstruction is an ultrashort pulse measurement technique.-The basics:...

(SPIDER).

## Polarization coherence

Light also has a polarization, which is the direction in which the electric field oscillates. Unpolarized light is composed of incoherent light waves with random polarization angles. The electric field of the unpolarized light wanders in every direction and changes in phase over the coherence time of the two light waves. An absorbing polarizerPolarizer

A polarizer is an optical filter that passes light of a specific polarization and blocks waves of other polarizations. It can convert a beam of light of undefined or mixed polarization into a beam with well-defined polarization. The common types of polarizers are linear polarizers and circular...

rotated to any angle will always transmit half the incident intensity when averaged over time.

If the electric field wanders by a smaller amount the light will be partially polarized so that at some angle, the polarizer will transmit more than half the intensity. If a wave is combined with an orthogonally polarized copy of itself delayed by less than the coherence time, partially polarized light is created.

The polarization of a light beam is represented by a vector in the Poincare sphere. For polarized light the end of the vector lies on the surface of the sphere, whereas the vector has zero length for unpolarized light. The vector for partially polarized light lies within the sphere

### Holography

Coherent superpositions of*optical wave fields*include holographyHolography

Holography is a technique that allows the light scattered from an object to be recorded and later reconstructed so that when an imaging system is placed in the reconstructed beam, an image of the object will be seen even when the object is no longer present...

. Holographic objects are used frequently in daily life in bank notes and credit cards.

### Non-optical wave fields

Further applications concern the coherent superposition of*non-optical wave fields*. In quantum mechanics for example one considers a probability field, which is related to the wave function (interpretation: density of the probability amplitude). Here the applications concern, among others, the future technologies of quantum computing and the already available technology of quantum cryptographyQuantum cryptography

Quantum key distribution uses quantum mechanics to guarantee secure communication. It enables two parties to produce a shared random secret key known only to them, which can then be used to encrypt and decrypt messages...

. Additionally the problems of the following subchapter are treated.

## Quantum coherence

In quantum mechanicsQuantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, all objects have wave-like properties (see de Broglie waves). For instance, in Young's double-slit experiment electrons can be used in the place of light waves. Each electron can go through either slit and hence has two paths that it can take to a particular final position. In quantum mechanics these two paths interfere. If there is destructive interference, the electron never arrives at that particular position. This ability to interfere is indicative of quantum coherence.

The quantum description of perfectly coherent paths is called a pure state, in which the two paths are combined in a superposition

Quantum superposition

Quantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system exists in all its particular, theoretically possible states simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations.Mathematically, it...

. The correlation between the two particles exceeds what would be predicted for classical correlation alone (see Bell's inequalities). If this two-particle system is decohered (which would occur in a measurement via Einselection

Einselection

Einselection is short for environment - induced superselection, a nickname coined by Wojciech H. Zurek. Classicality is an emergent property induced in open quantum systems by their environments...

), then there is no longer any phase relationship between the two states. The quantum description of imperfectly coherent paths is called a mixed state, described by a density matrix

Density matrix

In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...

(also called the "statistical operator") and analogous to a classical system of mixed probabilities. It has long been considered that, for mixed states, all correlations were entirely classical; however, in more recent work since 2001 it has been found that quantum correlations are present in certain mixed separable states and that such nonclassical correlations can be described within the conceptual framework of the so-called quantum discord

Quantum discord

In quantum information theory, quantum discord is a measure of nonclassical correlations between two subsystems of a quantum system. It includes correlations that are due to quantum physical effects but do not necessarily involve quantum entanglement....

.

Large-scale (macroscopic

Macroscopic

The macroscopic scale is the length scale on which objects or processes are of a size which is measurable and observable by the naked eye.When applied to phenomena and abstract objects, the macroscopic scale describes existence in the world as we perceive it, often in contrast to experiences or...

) quantum coherence leads to novel phenomena. For instance, the laser

Laser

A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

, superconductivity

Superconductivity

Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...

, and superfluidity are examples of highly coherent quantum systems, whose effects are evident at the macroscopic scale. These examples of quantum coherence are Bose–Einstein condensate

Bose–Einstein condensate

A Bose–Einstein condensate is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near absolute zero . Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at...

s. Here, all the particles that make up the condensate are in-phase; they are thus necessarily all described by a single quantum wavefunction.

On the other hand, the Schrödinger's cat

Schrödinger's cat

Schrödinger's cat is a thought experiment, usually described as a paradox, devised by Austrian physicist Erwin Schrödinger in 1935. It illustrates what he saw as the problem of the Copenhagen interpretation of quantum mechanics applied to everyday objects. The scenario presents a cat that might be...

thought experiment, highlights the fact that quantum coherence is not typically seen at the macroscopic scale but has been observed in the motion of a mechanical resonator (see Quantum machine

Quantum machine

A quantum machine is a human-made device whose collective motion follows the laws of quantum mechanics. The idea that macroscopic objects may follow the laws of quantum mechanics dates back to the advent of quantum mechanics in the early 20th century. However, as highlighted by the Schrödinger's...

).

## See also

- Atomic coherenceAtomic coherenceIn physics, atomic coherence is the induced coherence between levels of a multi-level atomic system sometimes observed when it interacts with a coherent electromagnetic field....
- Coherence lengthCoherence lengthIn physics, coherence length is the propagation distance from a coherent source to a point where an electromagnetic wave maintains a specified degree of coherence. The significance is that interference will be strong within a coherence length of the source, but not beyond it...
- Coherent stateCoherent stateIn quantum mechanics a coherent state is a specific kind of quantum state of the quantum harmonic oscillator whose dynamics most closely resembles the oscillating behaviour of a classical harmonic oscillator...
- Laser linewidthLaser linewidth←Laser linewidth is the spectral linewidth of a laser beam.Two of the most distinctive characteristics of laser emission are spatial coherence and spectral coherence. While spatial coherence is related to the beam divergence of the laser, spectral coherence is evaluated by measuring the laser...
- Measurement in quantum mechanicsMeasurement in quantum mechanicsThe framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus....
- Measurement problemMeasurement problemThe measurement problem in quantum mechanics is the unresolved problem of how wavefunction collapse occurs. The inability to observe this process directly has given rise to different interpretations of quantum mechanics, and poses a key set of questions that each interpretation must answer...
- Optical heterodyne detectionOptical heterodyne detectionOptical heterodyne detection is an important special case of heterodyne detection. In heterodyne detection, a signal of interest at some frequency is non-linearly mixed with a reference "local oscillator" that is set at a close-by frequency...
- Quantum decoherenceQuantum decoherenceIn quantum mechanics, quantum decoherence is the loss of coherence or ordering of the phase angles between the components of a system in a quantum superposition. A consequence of this dephasing leads to classical or probabilistically additive behavior...
- Quantum Zeno effectQuantum Zeno effectThe quantum Zeno effect is a name coined by George Sudarshan and Baidyanath Misra of the University of Texas in 1977 in their analysis of the situation in which an unstable particle, if observed continuously, will never decay. One can nearly "freeze" the evolution of the system by measuring it...