In quantum mechanics, the measurement problem is the problem of definite outcomes: quantum systems have superpositions but quantum measurements only give one definite result.
The wave function in quantum mechanics evolves deterministically according to the Schrödinger equation as a linear superposition
of different states. However, actual measurements always find the
physical system in a definite state. Any future evolution of the wave
function is based on the state the system was discovered to be in when
the measurement was made, meaning that the measurement "did something"
to the system that is not obviously a consequence of Schrödinger evolution.
The measurement problem is describing what that "something" is, how a
superposition of many possible values becomes a single measured value.
To express matters differently (paraphrasing Steven Weinberg),
the Schrödinger equation determines the wave function at any later
time. If observers and their measuring apparatus are themselves
described by a deterministic wave function, why can we not predict
precise results for measurements, but only probabilities? As a general
question: How can one establish a correspondence between quantum reality
and classical reality?
A thought experiment called Schrödinger's cat
illustrates the measurement problem. A mechanism is arranged to kill a
cat if a quantum event, such as the decay of a radioactive atom, occurs.
The mechanism and the cat are enclosed in a chamber so the fate of the
cat is unknown until the chamber is opened. Prior to observation,
according to quantum mechanics, the atom is in a quantum superposition, a linear combination
of decayed and intact states. Also according to quantum mechanics, the
atom-mechanism-cat composite system is described by superpositions of
compound states. Therefore, the cat would be described as in a
superposition, a linear combination of two states an "intact atom-alive
cat" and a "decayed atom-dead cat". However, when the chamber is opened
the cat is either alive or it is dead: there is no superposition
observed. After the measurement the cat is definitively alive or dead.[6]: 154
The cat scenario illustrates the measurement problem: how can an
indefinite superposition yield a single definite outcome? It also
illustrates other issues in quantum measurement,[7]: 585
including when does a measurement occur? Was it when the cat was
observed? How is a measurement apparatus defined? The mechanism for
detecting radioactive decay? The cat? The chamber? What the role of the
observer?
The views often grouped together as the Copenhagen interpretation are the oldest and, collectively, probably still the most widely held attitude about quantum mechanics.[8][9]N. David Mermin coined the phrase "Shut up and calculate!" to summarize Copenhagen-type views, a saying often misattributed to Richard Feynman and which Mermin later found insufficiently nuanced.[10][11]
Generally, views in the Copenhagen tradition posit something in the act of observation which results in the collapse of the wave function. This concept, though often attributed to Niels Bohr, was due to Werner Heisenberg, whose later writings obscured many disagreements he and Bohr had during their collaboration and that the two never resolved.[12][13]
In these schools of thought, wave functions may be regarded as
statistical information about a quantum system, and wave function
collapse is the updating of that information in response to new data.[14][15] Exactly how to understand this process remains a topic of dispute.[16]
Bohr offered an interpretation that is independent of a
subjective observer, or measurement, or collapse; instead, an
"irreversible" or effectively irreversible process causes the decay of
quantum coherence which imparts the classical behavior of "observation"
or "measurement".[17][18]
Hugh Everett's many-worlds interpretation
attempts to solve the problem by suggesting that there is only one wave
function, the superposition of the entire universe, and it never
collapses—so there is no measurement problem. Instead, the act of
measurement is simply an interaction between quantum entities, e.g.
observer, measuring instrument, electron/positron etc., which entangle
to form a single larger entity, for instance living cat/happy scientist. Everett also attempted to demonstrate how the probabilistic nature of quantum mechanics would appear in measurements, a work later extended by Bryce DeWitt.
However, proponents of the Everettian program have not yet reached a
consensus regarding the correct way to justify the use of the Born rule to calculate probabilities.[19][20]
The de Broglie–Bohm theory
tries to solve the measurement problem very differently: the
information describing the system contains not only the wave function,
but also supplementary data (a trajectory) giving the position of the
particle(s). The role of the wave function is to generate the velocity
field for the particles. These velocities are such that the probability
distribution for the particle remains consistent with the predictions of
the orthodox quantum mechanics. According to de Broglie–Bohm theory,
interaction with the environment during a measurement procedure
separates the wave packets in configuration space, which is where
apparent wave function collapse comes from, even though there is no
actual collapse.[21]
A fourth approach is given by objective-collapse models. In such models, the Schrödinger equation is modified and obtains nonlinear terms. These nonlinear modifications are of stochastic
nature and lead to behaviour that for microscopic quantum objects, e.g.
electrons or atoms, is unmeasurably close to that given by the usual
Schrödinger equation. For macroscopic objects, however, the nonlinear
modification becomes important and induces the collapse of the wave
function. Objective-collapse models are effective theories.
The stochastic modification is thought to stem from some external
non-quantum field, but the nature of this field is unknown. One possible
candidate is the gravitational interaction as in the models of Diósi
and Penrose. The main difference of objective-collapse models compared to the other approaches is that they make falsifiable
predictions that differ from standard quantum mechanics. Experiments
are already getting close to the parameter regime where these
predictions can be tested.[22]
The Ghirardi–Rimini–Weber (GRW) theory
proposes that wave function collapse happens spontaneously as part of
the dynamics. Particles have a non-zero probability of undergoing a
"hit", or spontaneous collapse of the wave function, on the order of
once every hundred million years.[23]
Though collapse is extremely rare, the sheer number of particles in a
measurement system means that the probability of a collapse occurring
somewhere in the system is high. Since the entire measurement system is
entangled (by quantum entanglement), the collapse of a single particle
initiates the collapse of the entire measurement apparatus. Because the
GRW theory makes different predictions from orthodox quantum mechanics
in some conditions, it is not an interpretation of quantum mechanics in a
strict sense.
The role of decoherence
Erich Joos and Heinz-Dieter Zeh claim that the phenomenon of quantum decoherence, which was put on firm ground in the 1980s, resolves the problem.[24]
The idea is that the environment causes the classical appearance of
macroscopic objects. Zeh further claims that decoherence makes it
possible to identify the fuzzy boundary between the quantum microworld
and the world where the classical intuition is applicable.[25][26] Quantum decoherence becomes an important part of some modern updates of the Copenhagen interpretation based on consistent histories.[27][28]
Quantum decoherence does not describe the actual collapse of the wave
function, but it explains the conversion of the quantum probabilities
(that exhibit interference effects) to the ordinary classical probabilities. See, for example, Zurek,[5] Zeh[25] and Schlosshauer.[29]
The present situation is slowly clarifying, described in a 2006 article by Schlosshauer as follows:[30]
Several decoherence-unrelated proposals have been put forward in the
past to elucidate the meaning of probabilities and arrive at the Born rule ... It is fair to say that no decisive conclusion appears to have been reached as to the success of these derivations. ...
As it is well known, [many papers by Bohr insist upon] the
fundamental role of classical concepts. The experimental evidence for
superpositions of macroscopically distinct states on increasingly large
length scales counters such a dictum. Superpositions appear to be novel
and individually existing states, often without any classical
counterparts. Only the physical interactions between systems then
determine a particular decomposition into classical states from the view
of each particular system. Thus classical concepts are to be understood
as locally emergent in a relative-state sense and should no longer
claim a fundamental role in the physical theory.
Six degrees of freedom (6DOF), or sometimes six degrees of movement, refers to the six mechanical degrees of freedom of movement of a rigid body in three-dimensional space. Specifically, the body is free to change position as forward/backward (surge), up/down (heave), left/right (sway) translation in three perpendicularaxes, combined with changes in orientation through rotation about three perpendicular axes, often termed yaw (normal axis), pitch (transverse axis), and roll (longitudinal axis).
Three degrees of freedom (3DOF), a term often used in the context of virtual reality, typically refers to tracking of rotational motion only: pitch, yaw, and roll.
Robotics
Serial and parallel manipulator systems are generally designed to position an end-effector with six degrees of freedom,
consisting of three in translation and three in orientation. This
provides a direct relationship between actuator positions and the
configuration of the manipulator defined by its forward and inverse kinematics.
Robot arms are described by their degrees of freedom.
This is a practical metric, in contrast to the abstract definition of
degrees of freedom which measures the aggregate positioning capability
of a system.
In 2007, Dean Kamen, inventor of the Segway, unveiled a prototype robotic arm with 14 degrees of freedom for DARPA. Humanoid robots
typically have 30 or more degrees of freedom, with six degrees of
freedom per arm, five or six in each leg, and several more in torso and neck.
Engineering
The term is important in mechanical systems, especially biomechanical systems,
for analyzing and measuring properties of these types of systems that
need to account for all six degrees of freedom. Measurement of the six
degrees of freedom is accomplished today through both AC and DC magnetic
or electromagnetic fields in sensors that transmit positional and
angular data to a processing unit. The data is made relevant through
software that integrates the data based on the needs and programming of
the users.
Mnemonics to remember angle names
The six degrees of freedom of a mobile unit are divided in two motional classes as described below.
Translational envelopes:
Moving forward and backward on the X-axis. (Surge)
There are three types of operational envelope in the Six degrees of freedom. These types are Direct, Semi-direct (conditional) and Non-direct,
all regardless of the time remaining for the execution of the maneuver,
the energy remaining to execute the maneuver and finally, if the motion
is commanded via a biological entity (e.g. human), a robotical entity
(e.g. computer) or both.
Direct type: Involved a degree can be commanded directly without particularly conditions and described as a normal operation. (An aileron on a basic airplane)
Semi-direct type: Involved a degree can be commanded when some specific conditions are met. (Reverse thrust on an aircraft)
Non-direct type: Involved a degree when is achieved via the interaction with its environment and cannot be commanded. (Pitching motion of a vessel at sea)
Transitional type also exists in some vehicles. For example, when the Space Shuttle operated in low Earth orbit, the craft was described as fully-direct-six because in the vacuum of space, its six degrees could be commanded via reaction wheels and RCS thrusters.
However, when the Space Shuttle was descending through the Earth's
atmosphere for its return, the fully-direct-six degrees were no longer
applicable as it was gliding through the air using its wings and control surfaces.
Game controllers
Six degrees of freedom also refers to movement in video game-play.
First-person shooter
(FPS) games generally provide five degrees of freedom:
forwards/backwards, slide left/right, up/down (jump/crouch/lie), yaw
(turn left/right), and pitch (look up/down). If the game allows leaning
control, then some consider it a sixth DOF; however, this may not be
completely accurate, as a lean is a limited partial rotation.
The term 6DOF has sometimes been used to describe games
which allow freedom of movement, but do not necessarily meet the full
6DOF criteria. For example, Dead Space 2, and to a lesser extent, Homeworld and Zone Of The Enders allow freedom of movement.
Motion tracking hardware devices such as TrackIR and software-based apps like Eyeware Beam are used for 6DOF head tracking. This device often finds its places in flight simulators and other vehicle simulators that require looking around the cockpit to locate enemies or simply avoiding accidents in-game.
The acronym 3DOF, meaning movement in the three dimensions but not rotation, is sometimes encountered.
The Razer Hydra, a motion controller for PC, tracks position and rotation of two wired nunchucks, providing six degrees of freedom on each hand.
The SpaceOrb 360 is a 6DOF computer input device released in 1996 originally manufactured and sold by the SpaceTec IMC company (first bought by Labtec, which itself was later bought by Logitech). They now offer the 3Dconnexion range of 6DOF controllers, primarily targeting the professional CAD industry.
The controllers sold with HTC VIVE provide 6DOF information by
the lighthouse, which adopts Time of Flight (TOF) technology to
determine the position of controllers.
In physics, coherence expresses the potential for two waves to interfere. Two monochromatic beams from a single source always interfere. Physical sources are not strictly monochromatic: they may be partly coherent. Beams from different sources are mutually incoherent.
When interfering, two waves add together to create a wave of greater amplitude than either one (constructive interference) or subtract from each other to create a wave of minima which may be zero (destructive interference), depending on their relative phase.
Constructive or destructive interference are limit cases, and two
waves always interfere, even if the result of the addition is
complicated or not remarkable.
Two waves with constant relative phase will be coherent. The amount of coherence can readily be measured by the interference visibility,
which looks at the size of the interference fringes relative to the
input waves (as the phase offset is varied); a precise mathematical
definition of the degree of coherence
is given by means of correlation functions. More generally, coherence
describes the statistical similarity of a field (electromagnetic field,
quantum wave packet etc.) at two points in space or time.
Qualitative concept
Two
slits illuminated by one source show an interference pattern. The
source is far to the left in the diagram, behind collimators that create
a parallel beam. This combination ensures that a wave from the source
strikes both slits at the same part of the wave cycle: the wave will
have coherence.
Coherence controls the visibility or contrast of interference patterns. For example, visibility of the double slit experiment pattern requires that both slits be illuminated by a coherent wave as illustrated in the figure. Large sources without collimation or sources that mix many different frequencies will have lower visibility.
Coherence contains several distinct concepts. Spatial coherence
describes the correlation (or predictable relationship) between waves
at different points in space, either lateral or longitudinal. Temporal coherence describes the correlation between waves observed at different moments in time. Both are observed in the Michelson–Morley experiment and Young's interference experiment. Once the fringes are obtained in the Michelson interferometer,
when one of the mirrors is moved away gradually from the beam-splitter,
the time for the beam to travel increases and the fringes become dull
and finally disappear, showing temporal coherence. Similarly, in a double-slit experiment,
if the space between the two slits is increased, the coherence dies
gradually and finally the fringes disappear, showing spatial coherence.
In both cases, the fringe amplitude slowly disappears, as the path
difference increases past the coherence length.
The coherence function between two signals and is defined as
where is the cross-spectral density of the signal and and are the power spectral density functions of and , respectively. The cross-spectral density and the power spectral density are defined as the Fourier transforms of the cross-correlation and the autocorrelation
signals, respectively. For instance, if the signals are functions of
time, the cross-correlation is a measure of the similarity of the two
signals as a function of the time lag relative to each other and the
autocorrelation is a measure of the similarity of each signal with
itself in different instants of time. In this case the coherence is a
function of frequency. Analogously, if and
are functions of space, the cross-correlation measures the similarity
of two signals in different points in space and the autocorrelations the
similarity of the signal relative to itself for a certain separation
distance. In that case, coherence is a function of wavenumber (spatial frequency).
The coherence varies in the interval . If it means that the signals are perfectly correlated or linearly related and if they are totally uncorrelated. If a linear system is being measured, being the input and
the output, the coherence function will be unitary all over the
spectrum. However, if non-linearities are present in the system the
coherence will vary in the limit given above.
Coherence and correlation
The coherence of two waves expresses how well correlated the waves are as quantified by the cross-correlation function.Cross-correlation quantifies the ability to predict the phase of the
second wave by knowing the phase of the first. As an example, consider
two waves perfectly correlated for all times (by using a monochromatic
light source). At any time, the phase difference between the two waves
will be constant. If, when they are combined, they exhibit perfect
constructive interference, perfect destructive interference, or
something in-between but with constant phase difference, 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 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 equation or some generalization thereof.
Waves in a rope (up and down) or slinky (compression and expansion)
In system with macroscopic waves, 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, one measures 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.
Temporal coherence
Figure 1: The amplitude of a single frequency wave as a function of time t (red) and a copy of the same wave delayed by (blue). The coherence time of the wave is infinite since it is perfectly correlated with itself for all delays .Figure 2: The amplitude of a wave whose phase drifts significantly in time as a function of time t (red) and a copy of the same wave delayed by (green). At any particular time t
the wave can interfere perfectly with its delayed copy. But, since half
the time the red and green waves are in phase and half the time out of
phase, when averaged over t any interference disappears at this delay.
Temporal coherence is the measure of the average correlation between the value of a wave and itself delayed by ,
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 time. At a delay of the degree of coherence is perfect, whereas it drops significantly as the delay passes . The coherence length is defined as the distance the wave travels in time .
The coherence time is not the time duration of the signal; the coherence length differs from the coherence area (see below).
The relationship between coherence time and bandwidth
The larger the bandwidth – range of frequencies Δf a wave contains – the faster the wave decorrelates (and hence the smaller is)
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.
Narrow bandwidth lasers have long coherence lengths (up to hundreds of meters). For example, a stabilized and monomode helium–neon laser can easily produce light with coherence lengths of 300 m. Not all lasers have a highly monochromaticity, 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.
Examples of temporal coherence
Examples of temporal coherence include:
A wave containing only a single frequency (monochromatic) is
perfectly correlated with itself at all time delays, in accordance with
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 packets)
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.
Holography requires light with a long coherence time. In contrast, optical coherence tomography, in its classical version, uses light with a short coherence time.
Measurement of temporal coherence
Figure 3: The amplitude of a wavepacket whose amplitude changes significantly in time (red) and a copy of the same wave delayed by (green) plotted as a function of time t.
At any particular time the red and green waves are uncorrelated; one
oscillates while the other is constant and so there will be no
interference at this delay. Another way of looking at this is the
wavepackets are not overlapped in time and so at any particular time
there is only one nonzero field so no interference can occur.Figure
4: The time-averaged intensity (blue) detected at the output of an
interferometer plotted as a function of delay τ for the example waves in
Figures 2 and 3. As the delay is changed by half a period, the
interference switches between constructive and destructive. The black
lines indicate the interference envelope, which gives the degree of coherence. Although the waves in Figures 2 and 3 have different time durations, they have the same coherence time.
In optics, temporal coherence is measured in an interferometer such as the Michelson interferometer or Mach–Zehnder interferometer. In these devices, a wave is combined with a copy of itself that is delayed by time . A detector measures the time-averaged intensity
of the light exiting the interferometer. The resulting visibility of
the interference pattern (e.g. see Figure 4) gives the temporal
coherence at delay .
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 , an infinitely fast detector would measure an intensity that fluctuates significantly over a time t equal to . In this case, to find the temporal coherence at , 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 spatial points x1 and x2
in the extent of a wave to interfere when averaged over time. More
precisely, the spatial coherence is the cross-correlation 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 defines the diameter of the coherence area, , (Coherence length ,
often a feature of a source, is usually an industrial term related to
the coherence time of the source, not the coherence area in the medium.)
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.
Figure 6: A wave with a varying profile (wavefront) and infinite coherence length.
Figure 7: A wave with a varying profile (wavefront) and finite coherence length.
Figure 8: A wave with finite coherence area is incident on a pinhole (small aperture). The wave will diffract
out of the pinhole. Far from the pinhole the emerging spherical
wavefronts are approximately flat. The coherence area is now infinite
while the coherence length is unchanged.
Figure 9: A wave with infinite
coherence area is combined with a spatially shifted copy of itself. Some
sections in the wave interfere constructively and some will interfere
destructively. Averaging over these sections, a detector with length D
will measure reduced interference visibility. For example, a misaligned Mach–Zehnder interferometer will do this.
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 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,
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 it was reported 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.Moreover, the coherence properties of the output light from multimode
nonlinear optical structures were found to obey the optical
thermodynamic theory.
Spectral coherence of short pulses
Figure 10: Waves of different frequencies interfere to form a localized pulse if they are coherent.Figure 11: Spectrally incoherent light interferes to form continuous light with a randomly varying phase and amplitude
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 transform).
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:
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 dispersion).
Light also has a polarization,
which is the direction in which the electric or magnetic 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 polarizer 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 Poincaré 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.
The signature property of quantum matter waves,
wave interference, relies on coherence. While initially patterned after
optical coherence, the theory and experimental understanding of quantum
coherence greatly expanded the topic.
Matter wave coherence
The simplest extension of optical coherence applies optical concepts to matter waves. For example, when performing the double-slit experiment
with atoms in place of light waves, a sufficiently collimated atomic
beam creates a coherent atomic wave-function illuminating both slits.
Each slit acts as a separate but in-phase beam contributing to the
intensity pattern on a screen. These two contributions give rise to an
intensity pattern of bright bands due to constructive interference,
interlaced with dark bands due to destructive interference, on a
downstream screen. Many variations of this experiment have been
demonstrated.
As with light, transverse coherence (across the direction of
propagation) of matter waves is controlled by collimation. Because
light, at all frequencies, travels the same velocity, longitudinal and
temporal coherence are linked; in matter waves these are independent. In
matter waves, velocity (energy) selection controls longitudinal
coherence and pulsing or chopping controls temporal coherence.
Quantum optics
The discovery of the Hanbury Brown and Twiss effect – correlation of light upon coincidence – triggered Glauber's creation
of uniquely quantum coherence analysis. Classical optical coherence
becomes a classical limit for first-order quantum coherence; higher
degree of coherence leads to many phenomena in quantum optics.
Macroscopic quantum coherence
Macroscopic scale quantum coherence leads to novel phenomena, the so-called macroscopic quantum phenomena. For instance, the laser, superconductivity and superfluidity
are examples of highly coherent quantum systems whose effects are
evident at the macroscopic scale. The macroscopic quantum coherence
(off-diagonal long-range order, ODLRO)
for superfluidity, and laser light, is related to first-order (1-body)
coherence/ODLRO, while superconductivity is related to second-order
coherence/ODLRO. (For fermions, such as electrons, only even orders of
coherence/ODLRO are possible.) For bosons, a Bose–Einstein condensate is an example of a system exhibiting macroscopic quantum coherence through a multiple occupied single-particle state.
The classical electromagnetic field exhibits macroscopic quantum
coherence. The most obvious example is the carrier signal for radio and
TV. They satisfy Glauber's quantum description of coherence.
Quantum coherence as a resource
Recently M. B. Plenio
and co-workers constructed an operational formulation of quantum
coherence as a resource theory. They introduced coherence monotones
analogous to the entanglement monotones. Quantum coherence has been shown to be equivalent to quantum entanglement
in the sense that coherence can be faithfully described as
entanglement, and conversely that each entanglement measure corresponds
to a coherence measure.
Applications
Holography
Coherent superpositions of optical wave fields include holography. Holographic photographs have been used as art and as difficult to forge security labels.
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 cryptography. Additionally the problems of the following subchapter are treated.
Modal analysis
Coherence
is used to check the quality of the transfer functions (FRFs) being
measured. Low coherence can be caused by poor signal to noise ratio,
and/or inadequate frequency resolution.