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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, qu ...
, a Fock state or number state is a
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 t ...
that is an element of a
Fock space The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after V. A. Fock who first intro ...
with a well-defined number of
particle In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...
s (or quanta). These states are named after the
Soviet The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen nationa ...
physicist
Vladimir Fock Vladimir Aleksandrovich Fock (or Fok; russian: Влади́мир Алекса́ндрович Фок) (December 22, 1898 – December 27, 1974) was a Soviet physicist, who did foundational work on quantum mechanics and quantum electrodynamic ...
. Fock states play an important role in the
second quantization Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as t ...
formulation of quantum mechanics. The particle representation was first treated in detail by
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
for
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 ...
s and by
Pascual Jordan Ernst Pascual Jordan (; 18 October 1902 – 31 July 1980) was a German theoretical and mathematical physicist who made significant contributions to quantum mechanics and quantum field theory. He contributed much to the mathematical form of matrix ...
and
Eugene Wigner Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his con ...
for
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s. The Fock states of bosons and fermions obey useful relations with respect to the Fock space
creation and annihilation operators Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually d ...
.


Definition

One specifies a multiparticle state of N non-interacting identical particles by writing the state as a sum of tensor products of N one-particle states. Additionally, depending on the integrality of the particles'
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
, the tensor products must be alternating (anti-symmetric) or symmetric products of the underlying one-particle
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. Specifically: *
Fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s, having half-integer spin and obeying the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
, correspond to antisymmetric tensor products. *
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 ...
s, possessing integer spin (and not governed by the exclusion principle) correspond to symmetric tensor products. If the number of particles is variable, one constructs the
Fock space The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after V. A. Fock who first intro ...
as the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of the tensor product Hilbert spaces for each
particle number The particle number (or number of particles) of a thermodynamic system, conventionally indicated with the letter ''N'', is the number of constituent particles in that system. The particle number is a fundamental parameter in thermodynamics which is ...
. In the Fock space, it is possible to specify the same state in a new notation, the occupancy number notation, by specifying the number of particles in each possible one-particle state. Let \left\_ be an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
of states in the underlying one-particle Hilbert space. This induces a corresponding basis of the Fock space called the "occupancy number basis". A quantum state in the Fock space is called a Fock state if it is an element of the occupancy number basis. A Fock state satisfies an important criterion: for each ''i'', the state is an eigenstate of the
particle number operator In quantum mechanics, for systems where the total number of particles may not be preserved, the number operator is the observable that counts the number of particles. The number operator acts on Fock space. Let :, \Psi\rangle_\nu=, \phi_1,\phi_ ...
\widehat corresponding to the ''i''-th elementary state ki. The corresponding eigenvalue gives the number of particles in the state. This criterion nearly defines the Fock states (one must in addition select a phase factor). A given Fock state is denoted by , n_,n_,..n_...\rangle. In this expression, n_ denotes the number of particles in the i-th state ki, and the particle number operator for the i-th state, \widehat, acts on the Fock state in the following way: : \widehat, n_,n_,..n_...\rangle = n_, n_,n_,..n_...\rangle Hence the Fock state is an eigenstate of the number operator with eigenvalue n_. Fock states often form the most convenient
basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of ...
of a Fock space. Elements of a Fock space that are superpositions of states of differing
particle number The particle number (or number of particles) of a thermodynamic system, conventionally indicated with the letter ''N'', is the number of constituent particles in that system. The particle number is a fundamental parameter in thermodynamics which is ...
(and thus not eigenstates of the number operator) are not Fock states. For this reason, not all elements of a Fock space are referred to as "Fock states". If we define the aggregate particle number operator \widehat as : \widehat = \sum_i \widehat, the definition of Fock state ensures that the
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
of measurement \operatorname\left(\widehat\right) = 0, i.e., measuring the number of particles in a Fock state always returns a definite value with no fluctuation.


Example using two particles

For any final state , f\rangle, any Fock state of two identical particles given by , 1_, 1_\rangle, and any operator \widehat , we have the following condition for indistinguishability: : \left, \left\langle f\left, \widehat\1_, 1_\right\rangle\^2 = \left, \left\langle f\left, \widehat\1_, 1_\right\rangle\^2 . So, we must have \left\langle f\left, \widehat\1_, 1_\right\rangle = e^\left\langle f\left, \widehat\1_, 1_\right\rangle where e^ = +1 for
bosons 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 ...
and -1 for
fermions In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
. Since \langle f, and \widehat are arbitrary, we can say, : \left, 1_, 1_\right\rangle = +\left, 1_, 1_\right\rangle for bosons and : \left, 1_, 1_\right\rangle = -\left, 1_, 1_\right\rangle for fermions. Note that the number operator does not distinguish bosons from fermions; indeed, it just counts particles without regard to their symmetry type. To perceive any difference between them, we need other operators, namely the
creation and annihilation operators Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually d ...
.


Bosonic Fock state

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 ...
s, which are particles with integer spin, follow a simple rule: their composite eigenstate is symmetric under operation by an
exchange operator In quantum mechanics, the exchange operator \hat, also known as permutation operator, is a quantum mechanical operator that acts on states in Fock space. The exchange operator acts by switching the labels on any two identical particles describ ...
. For example, in a two particle system in the tensor product representation we have \hat\left, x_1, x_2\right\rangle = \left, x_2, x_1\right\rangle .


Boson creation and annihilation operators

We should be able to express the same symmetric property in this new Fock space representation. For this we introduce non-Hermitian bosonic
creation and annihilation operators Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually d ...
, denoted by b^ and b respectively. The action of these operators on a Fock state are given by the following two equations: * Creation operator b^_ : *: b^_, n_, n_,n_...n_,...\rangle=\sqrt , n_, n_ ,n_...n_+1 ,...\rangle * Annihilation operator b_ : *: b_, n_, n_,n_...n_,...\rangle=\sqrt , n_, n_, n_...n_-1 ,...\rangle


Non-Hermiticity of creation and annihilation operators

The bosonic Fock state creation and annihilation operators are not
Hermitian operators In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to its ...
.


Operator identities

The commutation relations of creation and annihilation operators in a bosonic system are : \left ^_i, b^\dagger_j\right\equiv b^_i b^\dagger_j - b^\dagger_jb^_i = \delta_, : \left ^\dagger_i, b^\dagger_j\right= \left ^_i, b^_j\right= 0, where \ , \ \ /math> is the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
and \delta_ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
.


N bosonic basis states , n_, n_ ,n_...n_,...\rangle


Action on some specific Fock states


Action of number operators

The number operators \widehat for a bosonic system are given by \widehat=b^_b_, where \widehat, n_, n_ ,n_...n_...\rangle=n_ , n_, n_ ,n_...n_...\rangle Number operators are Hermitian operators.


Symmetric behaviour of bosonic Fock states

The commutation relations of the creation and annihilation operators ensure that the bosonic Fock states have the appropriate symmetric behaviour under particle exchange. Here, exchange of particles between two states (say, ''l'' and ''m'') is done by annihilating a particle in state ''l'' and creating one in state ''m''. If we start with a Fock state , \psi\rangle = \left, n_, n_ , .... n_ ... n_ ... \right\rangle, and want to shift a particle from state k_l to state k_m, then we operate the Fock state by b_^\dagger b_ in the following way: Using the commutation relation we have, b_^\dagger.b_ = b_.b_^\dagger : \begin b_^\dagger.b_ \left, n_, n_, .... n_ ... n_ ... \right\rangle &= b_.b_^\dagger \left, n_, n_, .... n_ ... n_ ... \right\rangle \\ &= \sqrt\sqrt \left, n_, n_, .... n_ + 1 ... n_ - 1 ...\right\rangle \end So, the Bosonic Fock state behaves to be symmetric under operation by Exchange operator. Wignerfunction fock 0.png, Wigner function of , 0\rangle Wignerfunction fock 1.png, Wigner function of , 1\rangle Wignerfunction fock 2.png, Wigner function of , 2\rangle Wignerfunction fock 3.png, Wigner function of , 3\rangle Wignerfunction fock 4.png, Wigner function of , 4\rangle


Fermionic Fock state


Fermion creation and annihilation operators

To be able to retain the antisymmetric behaviour of
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s, for Fermionic Fock states we introduce non-Hermitian fermion creation and annihilation operators, defined for a Fermionic Fock state , \psi\rangle = , n_, n_ ,n_...n_,...\rangle as: * The creation operator c^_ acts as: *: c^_, n_, n_ ,n_...n_,...\rangle=\sqrt , n_, n_ ,n_...n_+1 ,...\rangle * The annihilation operator c_ acts as: *: c_, n_, n_ ,n_...n_,...\rangle=\sqrt , n_, n_ ,n_...n_-1 ,...\rangle These two actions are done antisymmetrically, which we shall discuss later.


Operator identities

The anticommutation relations of creation and annihilation operators in a fermionic system are, : \begin \left\ \equiv c^_i c^\dagger_j + c^\dagger_jc^_i &= \delta_, \\ \left\ = \left\ &= 0, \end where is the
anticommutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
and \delta_ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
. These anticommutation relations can be used to show antisymmetric behaviour of ''Fermionic Fock states''.


Action of number operators

Number operators \widehat for
Fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s are given by \widehat=c^_.c_. : \widehat, n_, n_ ,n_...n_...\rangle=n_ , n_, n_ ,n_...n_...\rangle


Maximum occupation number

The action of the number operator as well as the creation and annihilation operators might seem same as the bosonic ones, but the real twist comes from the maximum occupation number of each state in the fermionic Fock state. Extending the 2-particle fermionic example above, we first must convince ourselves that a fermionic Fock state , \psi\rangle = \left, n_, n_, n_ ... n_ ... \right\rangle is obtained by applying a certain sum of permutation operators to the tensor product of eigenkets as follows: : \left, n_, n_, n_ ... n_ ...\right\rangle = S_-\left, i_1, i_2, i_3 ... i_l ...\right\rangle = \frac\begin \left, i_1\right\rangle_1 & \cdots & \left, i_1\right\rangle_N \\ \vdots & \ddots & \vdots \\ \left, i_N\right\rangle_1 & \cdots & \left, i_N\right\rangle_N \end This determinant is called the
Slater determinant In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electr ...
. If any of the single particle states are the same, two rows of the Slater determinant would be the same and hence the determinant would be zero. Hence, two identical
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s must not occupy the same state (a statement of the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
). Therefore, the occupation number of any single state is either 0 or 1. The eigenvalue associated to the fermionic Fock state \widehat must be either 0 or 1.


N fermionic basis states \left, n_, n_, n_ ... n_, ...\right\rangle


Action on some specific Fock states


Antisymmetric behaviour of Fermionic Fock state

Antisymmetric behaviour of Fermionic states under Exchange operator is taken care of the anticommutation relations. Here, exchange of particles between two states is done by annihilating one particle in one state and creating one in other. If we start with a Fock state , \psi\rangle = \left, n_, n_, ... n_... n_ ...\right\rangle and want to shift a particle from state k_l to state k_m, then we operate the Fock state by c_^.c_ in the following way: Using the anticommutation relation we have : c_^\dagger.c_ = -c_.c_^\dagger : c_^.c_ \left, n_, n_, .... n_ ... n_ ... \right\rangle = \sqrt\sqrt \left, n_, n_, .... n_ + 1 ... n_ - 1 ...\right\rangle but, \begin &c_.c_^, n_, n_, ....n_... n_...\rangle \\ = -&c_^.c_, n_, n_, .... n_... n_...\rangle \\ = -&\sqrt\sqrt, n_, n_, .... n_ + 1 ... n_ - 1...\rangle \end Thus, fermionic Fock states are antisymmetric under operation by particle exchange operators.


Fock states are not energy eigenstates in general

In
second quantization Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as t ...
theory, the Hamiltonian density function is given by :\mathfrak = \frac \nabla_\psi^(x)\, \nabla_\psi(x) The total
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
is given by :\begin \mathcal &= \int d^3 x\,\mathfrak = \int d^x \psi^(x)\left(-\frac\right)\psi(x) \\ \therefore \mathfrak &= -\frac \end In free Schrödinger theory, :\mathfrak\psi_^(x) = -\frac\psi_^(x) = E_^\psi_^(x) and :\int d^3 x\, \psi_^(x)\, \psi_^(x) = \delta_ and : \psi(x) = \sum_n a_n \psi_^(x), where a_n is the annihilation operator. : \therefore \mathcal = \sum_\int d^x\, a^_\psi_^(x)\, \mathfraka_n \psi_^(x) Only for non-interacting particles do \mathfrak and a_n commute; in general they do not commute. For non-interacting particles, : \mathcal = \sum_\int d^3 x\, a^_\psi_^(x)\, E^_\psi_^(x)a_n = \sum_E^_ a^_a_n\delta_ = \sum_E^_a^_n a_n = \sum_E^_\widehat If they do not commute, the Hamiltonian will not have the above expression. Therefore, in general, Fock states are not energy eigenstates of a system.


Vacuum fluctuations

The vacuum state or , 0\rangle is the state of lowest energy and the expectation values of a and a^\dagger vanish in this state: :a, 0\rangle = 0 = \langle 0 , a^\dagger The electrical and magnetic fields and the vector potential have the mode expansion of the same general form: :F\left(\vec, t\right) = \varepsilon a e^ + h\cdot c Thus it is easy to see that the expectation values of these field operators vanishes in the vacuum state: :\langle 0, F, 0 \rangle = 0 However, it can be shown that the expectation values of the square of these field operators is non-zero. Thus there are fluctuations in the field about the zero ensemble average. These vacuum fluctuations are responsible for many interesting phenomenon including the
Lamb shift In physics, the Lamb shift, named after Willis Lamb, is a difference in energy between two energy levels 2''S''1/2 and 2''P''1/2 (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which th ...
in quantum optics.


Multi-mode Fock states

In a multi-mode field each creation and annihilation operator operates on its own mode. So a_ and a^_ will operate only on \left, n_\right\rangle. Since operators corresponding to different modes operate in different sub-spaces of the Hilbert space, the entire field is a direct product of , n_\rangle over all the modes: : \left, n_\right\rangle \left, n_\right\rangle \left, n_\right\rangle \ldots \equiv \left, n_, n_, n_... n_... \right\rangle \equiv \left, \\right\rangle The creation and annihilation operators operate on the multi-mode state by only raising or lowering the number state of their own mode: :\begin a_ , n_, n_, n_... n_, ...\rangle &= \sqrt , n_, n_, n_... n_-1, ...\rangle \\ a^_ , n_, n_, n_... n_,...\rangle &= \sqrt , n_, n_, n_... n_ + 1, ...\rangle \end We also define the total
number operator In quantum mechanics, for systems where the total number of particles may not be preserved, the number operator is the observable that counts the number of particles. The number operator acts on Fock space. Let :, \Psi\rangle_\nu=, \phi_1,\phi_2 ...
for the field which is a sum of number operators of each mode: : \hat_ = \sum \hat_ The multi-mode Fock state is an eigenvector of the total number operator whose eigenvalue is the total occupation number of all the modes : \hat_ , \\rangle = \left( \sum n_ \right) , \\rangle In case of non-interacting particles, number operator and Hamiltonian commute with each other and hence multi-mode Fock states become eigenstates of the multi-mode Hamiltonian : \hat \left, \\right\rangle = \left( \sum \hbar \omega \left(n_ + \frac \right)\right) \left, \\right\rangle


Source of single photon state

Single photons are routinely generated using single emitters (atoms,
Nitrogen-vacancy center The nitrogen-vacancy center (N-V center or NV center) is one of numerous point defects in diamond. Its most explored and useful property is its photoluminescence, which allows observers to read out its spin-state. The NV center's electron spin, lo ...
,
Quantum dot Quantum dots (QDs) are semiconductor particles a few nanometres in size, having optical and electronic properties that differ from those of larger particles as a result of quantum mechanics. They are a central topic in nanotechnology. When the ...
C. Santori, M. Pelton, G. Solomon, Y. Dale and Y. Yamamoto (2001), "Triggered Single Photons from a Quantum Dot", ''Phys. Rev. Lett.'' 86 (8):1502--150
DOI 10.1103/PhysRevLett.86.1502
). However, these sources are not always very efficient, often presenting a low probability of actually getting a single photon on demand; and often complex and unsuitable out of a laboratory environment. Other sources are commonly used that overcome these issues at the expense of a nondeterministic behavior. Heralded single photon sources are probabilistic two-photon sources from whom the pair is split and the detection of one photon heralds the presence of the remaining one. These sources usually rely on the optical nonlinearity of some materials like periodically poled
Lithium niobate Lithium niobate () is a non-naturally-occurring salt consisting of niobium, lithium, and oxygen. Its single crystals are an important material for optical waveguides, mobile phones, piezoelectric sensors, optical modulators and various other linea ...
(
Spontaneous parametric down-conversion Spontaneous parametric down-conversion (also known as SPDC, parametric fluorescence or parametric scattering) is a nonlinear instant optical process that converts one photon of higher energy (namely, a pump photon), into a pair of photons (namely, ...
), or silicon (spontaneous
Four-wave mixing Four-wave mixing (FWM) is an intermodulation phenomenon in nonlinear optics, whereby interactions between two or three wavelengths produce two or one new wavelengths. It is similar to the third-order intercept point in electrical systems. Four-wave ...
) for example.


Non-classical behaviour

The Glauber–Sudarshan P-representation of Fock states shows that these states are purely quantum mechanical and have no classical counterpart. The \scriptstyle\varphi(\alpha) \, of these states in the representation is a 2n'th derivative of the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
and therefore not a classical probability distribution.


See also

*
Coherent states In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harmo ...
* Heisenberg limit * Nonclassical light


References

{{reflist, 2


External links

*Vladan Vuletic of
MIT The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the m ...
ha
used an ensemble of atoms to produce a Fock state (a.k.a. single photon) source
(PDF) * Produce and measure a single photon state (Fock state) with an interactive experimen
QuantumLab
Quantum optics Quantum field theory