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 ...
,
correlation functions are used to characterize the statistical and
coherence
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 deriv ...
properties of an electromagnetic field. The degree of coherence is the normalized correlation of electric fields; in its simplest form, termed
. It is useful for quantifying the coherence between two electric fields, as measured in a Michelson or other linear optical
interferometer
Interferometry is a technique which uses the '' interference'' of superimposed waves to extract information. Interferometry typically uses electromagnetic waves and is an important investigative technique in the fields of astronomy, fiber o ...
. The correlation between pairs of fields,
, typically is used to find the statistical character of intensity fluctuations. First order correlation is actually the amplitude-amplitude correlation and the second order correlation is the intensity-intensity correlation. It is also used to differentiate between states of light that require a
quantum mechanical description and those for which classical fields are sufficient. Analogous considerations apply to any Bose field in subatomic physics, in particular to mesons (cf.
Bose–Einstein correlations In physics, Bose–Einstein correlations are correlations between identical bosons. They have important applications in astronomy, optics, particle and nuclear physics.
From intensity interferometry to Bose–Einstein correlations
The interf ...
).
Degree of first-order coherence
The normalized first order correlation function is written as:
:
where
denotes a (statistical) ensemble average. For non-stationary states, such as pulses, the ensemble is made up of many pulses. When one deals with stationary states, where the statistical properties do not change with time, one can replace the ensemble average with a time average. If we restrict ourselves to plane parallel to each other waves then
.
In this case, the result for stationary states will not depend on
, but on the time delay
(or
if
).
This allows us to write a simplified form
:
where we have now averaged over ''t''.
Applications
In optical interferometers such as the
Michelson interferometer
The Michelson interferometer is a common configuration for optical interferometry and was invented by the 19/20th-century American physicist Albert Abraham Michelson. Using a beam splitter, a light source is split into two arms. Each of those ...
,
Mach–Zehnder interferometer
The Mach–Zehnder interferometer is a device used to determine the relative phase shift variations between two collimated beams derived by splitting light from a single source. The interferometer has been used, among other things, to measure p ...
, or
Sagnac interferometer
The Sagnac effect, also called Sagnac interference, named after French physicist Georges Sagnac, is a phenomenon encountered in interferometry that is elicited by rotation. The Sagnac effect manifests itself in a setup called a ring interferomete ...
, one splits an electric field into two components, introduces a time delay to one of the components, and then recombines them. The intensity of resulting field is measured as a function of the time delay. In this specific case involving two equal input intensities, the
visibility
The visibility is the measure of the distance at which an object or light can be clearly discerned. In meteorology it depends on the transparency of the surrounding air and as such, it is unchanging no matter the ambient light level or time o ...
of the resulting interference pattern is given by:
:
where the second expression involves combining two space-time points from a field.
The visibility ranges from zero, for incoherent electric fields, to one, for coherent electric fields. Anything in between is described as partially coherent.
Generally,
and
.
Examples of ''g''(1)
For light of a single frequency (of a point source):
:
For Lorentzian
chaotic light (e.g. collision broadened):
:
For Gaussian
chaotic light (e.g. Doppler broadened):
:
Here,
is the central frequency of the light and
is the
coherence time
For an electromagnetic wave, the coherence time is the time over which a propagating wave (especially a laser or maser beam) may be considered coherent, meaning that its phase is, on average, predictable.
In long-distance transmission systems ...
of the light.
Degree of second-order coherence
The normalised second order correlation function is written as:
:
Note that this is not a generalization of the first-order coherence
If the electric fields are considered classical, we can reorder them to express
in terms of intensities. A plane parallel wave in a stationary state will have
:
The above expression is even,
. For classical fields, one can apply the
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics.
The inequality for sums was published by . The corresponding inequality f ...
to the intensities in the above expression (since they are real numbers) to show that
. The inequality
shows that
. Assuming independence of intensities when
leads to
. Nevertheless, the second-order coherence for an average over fringes of complementary
interferometer
Interferometry is a technique which uses the '' interference'' of superimposed waves to extract information. Interferometry typically uses electromagnetic waves and is an important investigative technique in the fields of astronomy, fiber o ...
outputs of a coherent state is only 0.5 (even though
for each output). And
(calculated from averages) can be reduced down to zero with a proper discriminating
trigger level applied to the signal (within the range of coherence).
Examples of ''g''(2)
* Chaotic light of all kinds:
.
Note the
Hanbury Brown and Twiss effect uses this fact to find
from a measurement of
.
* Light of a single frequency:
.
* In the case of
photon antibunching
Photon antibunching generally refers to a light field with photons more equally spaced than a coherent laser field, a signature being signals at appropriate detectors which are anticorrelated. More specifically, it can refer to sub-Poissonian p ...
, for
we have
for a single photon source because
*:
*: where
is the photon number observable.
[SINGLE PHOTONS FOR QUANTUM INFORMATION
PROCESSING - http://www.stanford.edu/group/yamamotogroup/Thesis/DFthesis.pdf (Archived copy: https://web.archive.org/web/20121023140645/http://www.stanford.edu/group/yamamotogroup/Thesis/DFthesis.pdf)]
Degree of ''n''th-order coherence
A generalization of the first-order coherence
:
A generalization of the second-order coherence
:
or in intensities
:
Examples of ''g''(''n'')
Light of a single frequency:
:
Using the first definition:
Chaotic light of all kinds:
Using the second definition:
Chaotic light of all kinds:
Chaotic light of all kinds:
Generalization to quantum fields
The predictions of
for ''n'' > 1 change when the classical fields (
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
or
c-number
The term Number C (or C number) is an old nomenclature used by Paul Dirac which refers to real and complex numbers. It is used to distinguish from operators (q-numbers or quantum numbers) in quantum mechanics.
Although c-numbers are commuting, th ...
s) are replaced with quantum fields (operators or
q-number
In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be k ...
s). In general, quantum fields do not necessarily commute, with the consequence that their order in the above expressions can not be simply interchanged.
:
With
:
we get in the case of stationary light:
:
Photon bunching
Light is said to be bunched if
and antibunched if
.
See also
*
Bose–Einstein correlations In physics, Bose–Einstein correlations are correlations between identical bosons. They have important applications in astronomy, optics, particle and nuclear physics.
From intensity interferometry to Bose–Einstein correlations
The interf ...
*
Coherence theory
*
Correlation and dependence
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistic ...
*
Fourier transform spectroscopy
Fourier-transform spectroscopy is a measurement technique whereby spectra are collected based on measurements of the coherence of a radiative source, using time-domain or space-domain measurements of the radiation, electromagnetic or not. It c ...
*
*
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 com ...
*
Orders Of Coherence Coherence is defined as the ability of waves to interfere. Intuitively, coherent waves have a well-defined constant phase relationship. However, an exclusive and extensive physical definition of coherence is more nuanced. Coherence functions, as int ...
*
Van Cittert–Zernike theorem
The van Cittert–Zernike theorem, named after physicists Pieter Hendrik van Cittert and Frits Zernike, is a formula in coherence theory that states that under certain conditions the Fourier transform of the intensity distribution function of a ...
*
Interferometric visibility The interferometric visibility (also known as interference visibility and fringe visibility, or just visibility when in context) is a measure of the contrast of '' interference'' in any system subject to wave superposition.
Examples include as opt ...
References
Suggested reading
* Loudon, Rodney, ''The Quantum Theory of Light'' (Oxford University Press, 2000), {{ISBN, 0-19-850177-3
Quantum optics