Inquantum mechanics, aFock state ornumber state is aquantum state that is an element of aFock space with a well-defined number ofparticles (orquanta). These states are named after theSoviet physicistVladimir Fock. Fock states play an important role in thesecond quantization formulation of quantum mechanics.

The particle representation was first treated in detail byPaul Dirac forbosons and byPascual Jordan andEugene Wigner forfermions.^[1]: 35 The Fock states of bosons and fermions obey useful relations with respect to the Fock spacecreation and annihilation operators.

One specifies a multiparticle state ofN non-interacting identical particles by writing the state as a sum oftensor products ofN one-particle states. Additionally, depending on the integrality of the particles'spin, the tensor products must bealternating (anti-symmetric) orsymmetric products of the underlying one-particleHilbert spaces. Specifically:

• Fermions, havinghalf-integer spin and obeying thePauli exclusion principle, correspond to antisymmetric tensor products.
• Bosons, possessing integer spin (and not governed by the exclusion principle) correspond to symmetric tensor products.

If the number of particles is variable, one constructs theFock space as thedirect sum of the tensor product Hilbert spaces for eachparticle number. 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${\left\{{\mathbf{k}}_i\right\}}_{i\in I}$ be anorthonormal basis 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 aFock state if it is an element of the occupancy number basis.

A Fock state satisfies an important criterion: for eachi, the state is an eigenstate of theparticle number operator${\displaystyle \hat{N_{{\mathbf{k}}_i}}}$ corresponding to thei-th elementary statek_i. The correspondingeigenvalue gives the number of particles in the state. This criterion nearly defines the Fock states (one must in addition select aphase factor).

A given Fock state is denoted by${\displaystyle |n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},..n_{{\mathbf{k}}_i}...⟩}$. In this expression,${\displaystyle n_{{\mathbf{k}}_i}}$ denotes the number of particles in the i-th statek_i, and the particle number operator for the i-th state,${\displaystyle \hat{N_{{\mathbf{k}}_i}}}$, acts on the Fock state in the following way:

${\displaystyle \hat{N_{{\mathbf{k}}_i}}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},..n_{{\mathbf{k}}_i}...⟩=n_{{\mathbf{k}}_i}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},..n_{{\mathbf{k}}_i}...⟩}$

Hence the Fock state is an eigenstate of the number operator with eigenvalue${\displaystyle n_{{\mathbf{k}}_i}}$.^[2]: 478

Fock states often form the most convenientbasis of a Fock space. Elements of a Fock space that aresuperpositions of states of differingparticle number (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$\hat{N}$ as

${\displaystyle \hat{N}=\sum _i\hat{N_{{\mathbf{k}}_i}},}$

the definition of Fock state ensures that thevariance of measurement${\displaystyle \mathrm{Var}\left(\hat{N}\right)=0}$, i.e., measuring the number of particles in a Fock state always returns a definite value with no fluctuation.

For any final state${\displaystyle |f⟩}$, any Fock state of two identical particles given by${\displaystyle |1_{{\mathbf{k}}_1},1_{{\mathbf{k}}_2}⟩}$, and anyoperator${\displaystyle \hat{\mathbb{O}}}$, we have the following condition forindistinguishability:^[3]: 191

${\displaystyle {\left|⟨f\left|\hat{\mathbb{O}}\right|1_{{\mathbf{k}}_1},1_{{\mathbf{k}}_2}⟩\right|}^2={\left|⟨f\left|\hat{\mathbb{O}}\right|1_{{\mathbf{k}}_2},1_{{\mathbf{k}}_1}⟩\right|}^2}$.

So, we must have${\displaystyle ⟨f\left|\hat{\mathbb{O}}\right|1_{{\mathbf{k}}_1},1_{{\mathbf{k}}_2}⟩=e^{i\delta }⟨f\left|\hat{\mathbb{O}}\right|1_{{\mathbf{k}}_2},1_{{\mathbf{k}}_1}⟩}$

where${\displaystyle e^{i\delta }=+1}$ forbosons and${\displaystyle -1}$ forfermions. Since${\displaystyle ⟨f|}$ and${\displaystyle \hat{\mathbb{O}}}$ are arbitrary, we can say,

${\displaystyle |1_{{\mathbf{k}}_1},1_{{\mathbf{k}}_2}⟩=+|1_{{\mathbf{k}}_2},1_{{\mathbf{k}}_1}⟩}$ for bosons and

${\displaystyle |1_{{\mathbf{k}}_1},1_{{\mathbf{k}}_2}⟩=-|1_{{\mathbf{k}}_2},1_{{\mathbf{k}}_1}⟩}$ for fermions.^[3]: 191

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 thecreation and annihilation operators.

Bosons, which are particles with integer spin, follow a simple rule: their composite eigenstate is symmetric^[4] under operation by anexchange operator. For example, in a twoparticle system in the tensor product representation we have${\displaystyle \hat{P}|x_1,x_2⟩=|x_2,x_1⟩}$ .

We should be able to express the same symmetric property in this new Fock space representation. For this we introduce non-Hermitian bosoniccreation and annihilation operators,^[4] denoted by${\displaystyle b^{†}}$ and${\displaystyle b}$ respectively. The action of these operators on a Fock state are given by the following two equations:

• Creation operator$b_{{\mathbf{k}}_l}^{†}$:

  ${\displaystyle b_{{\mathbf{k}}_l}^{†}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},n_{{\mathbf{k}}_3}...n_{{\mathbf{k}}_l},...⟩=\sqrt{n_{{\mathbf{k}}_l}+1}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},n_{{\mathbf{k}}_3}...n_{{\mathbf{k}}_l}+1,...⟩}$^[4]

• Annihilation operator$b_{{\mathbf{k}}_l}$:

  ${\displaystyle b_{{\mathbf{k}}_l}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},n_{{\mathbf{k}}_3}...n_{{\mathbf{k}}_l},...⟩=\sqrt{n_{{\mathbf{k}}_l}}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},n_{{\mathbf{k}}_3}...n_{{\mathbf{k}}_l}-1,...⟩}$^[4]

The bosonic Fock state creation and annihilation operators are notHermitian operators.^[4]

The commutation relations of creation and annihilation operators in abosonic system are

${\displaystyle \left[b_i^{},b_j^{†}\right]\equiv b_i^{}{b}_j^{†}-b_j^{†}{b}_i^{}={\delta }_{ij},}$^[4]

${\displaystyle \left[b_i^{†},b_j^{†}\right]=\left[b_i^{},b_j^{}\right]=0,}$^[4]

where${\displaystyle [,]}$ is thecommutator and${\displaystyle {\delta }_{ij}}$ is theKronecker delta.

Number of particles (N)	Bosonic basis states[6]: 11
0	${\displaystyle |0,0,\mathrm{0...}⟩}$
1	${\displaystyle |1,0,\mathrm{0...}⟩}$,${\displaystyle |0,1,\mathrm{0...}⟩}$,${\displaystyle |0,0,\mathrm{1...}⟩}$,...
2	${\displaystyle |2,0,\mathrm{0...}⟩}$,${\displaystyle |1,1,\mathrm{0...}⟩}$,${\displaystyle |0,2,\mathrm{0...}⟩}$,...
${\displaystyle n}$	${\displaystyle |n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},n_{{\mathbf{k}}_3}...n_{{\mathbf{k}}_l},...⟩}$

The number operators$\hat{N_{{\mathbf{k}}_l}}$ for a bosonic system are given by${\displaystyle \hat{N_{{\mathbf{k}}_l}}=b_{{\mathbf{k}}_l}^{†}{b}_{{\mathbf{k}}_l}}$, where${\displaystyle \hat{N_{{\mathbf{k}}_l}}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},n_{{\mathbf{k}}_3}...n_{{\mathbf{k}}_l}...⟩=n_{{\mathbf{k}}_l}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},n_{{\mathbf{k}}_3}...n_{{\mathbf{k}}_l}...⟩}$^[4]

Number operators are Hermitian operators.

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 andm) is done by annihilating a particle in statel and creating one in statem. If we start with a Fock state${\displaystyle |\psi ⟩=|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},....n_{{\mathbf{k}}_m}...n_{{\mathbf{k}}_l}...⟩}$, and want to shift a particle from state${\displaystyle k_l}$ to state${\displaystyle k_m}$, then we operate the Fock state by${\displaystyle b_{{\mathbf{k}}_m}^{†}{b}_{{\mathbf{k}}_l}}$ in the following way:

Using the commutation relation we have,${\displaystyle b_{{\mathbf{k}}_m}^{†}.b_{{\mathbf{k}}_l}=b_{{\mathbf{k}}_l}.b_{{\mathbf{k}}_m}^{†}}$

So, the Bosonic Fock state behaves to be symmetric under operation by Exchange operator.

• 
  Wigner function of${\displaystyle |0⟩}$
• 
  Wigner function of${\displaystyle |1⟩}$
• 
  Wigner function of${\displaystyle |2⟩}$
• 
  Wigner function of${\displaystyle |3⟩}$
• 
  Wigner function of${\displaystyle |4⟩}$

In the occupation number representation the single particle basis states are written with the occupation number of each orbital. For Fermion states, the occupation number can only be either zero or one and the order of the orbitals is significant.^[6]: 10

Number of particles (N)	Fermionic basis states[6]: 11  ${\displaystyle |n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},n_{{\mathbf{k}}_3}...n_{{\mathbf{k}}_l},...⟩}$
0	${\displaystyle |0,0,\mathrm{0...}⟩}$
1	${\displaystyle |1,0,\mathrm{0...}⟩}$,${\displaystyle |0,1,\mathrm{0...}⟩}$,${\displaystyle |0,0,\mathrm{1...}⟩}$,...
2	${\displaystyle |1,1,\mathrm{0...}⟩}$,${\displaystyle |0,1,\mathrm{1...}⟩}$,${\displaystyle |0,1,0,\mathrm{1...}⟩}$,${\displaystyle |1,0,1,\mathrm{0...}⟩}$...
...	...

To retain the antisymmetric behaviour offermions non-Hermitian fermion creation and annihilation operators are defined for a Fermionic Fock state${\displaystyle |\psi ⟩=|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},n_{{\mathbf{k}}_3}...n_{{\mathbf{k}}_l},...⟩}$ as:^[4]

• The creation operator${\displaystyle c_{{\mathbf{k}}_l}^{†}}$ acts on the basis states as:

  ${\displaystyle c_{{\mathbf{k}}_l}^{†}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},n_{{\mathbf{k}}_3}...n_{{\mathbf{k}}_l},...⟩=\sqrt{n_{{\mathbf{k}}_l}+1}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},n_{{\mathbf{k}}_3}...n_{{\mathbf{k}}_l}+1,...⟩}$^[4]

• The annihilation operator$c_{{\mathbf{k}}_l}$ acts as:

  ${\displaystyle c_{{\mathbf{k}}_l}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},n_{{\mathbf{k}}_3}...n_{{\mathbf{k}}_l},...⟩=\sqrt{n_{{\mathbf{k}}_l}}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},n_{{\mathbf{k}}_3}...n_{{\mathbf{k}}_l}-1,...⟩}$

The anticommutation relations of creation and annihilation operators in afermionic system are,

^[4]

where${\displaystyle \{,\}}$ is theanticommutator and${\displaystyle {\delta }_{ij}}$ is theKronecker delta. These anticommutation relations can be used to show antisymmetric behaviour ofFermionic Fock states.

Number operators$\hat{N_{{\mathbf{k}}_l}}$ forFermions are given by${\displaystyle \hat{N_{{\mathbf{k}}_l}}=c_{{\mathbf{k}}_l}^{†}.c_{{\mathbf{k}}_l}}$.

${\displaystyle \hat{N_{{\mathbf{k}}_l}}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},n_{{\mathbf{k}}_3}...n_{{\mathbf{k}}_l}...⟩=n_{{\mathbf{k}}_l}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},n_{{\mathbf{k}}_3}...n_{{\mathbf{k}}_l}...⟩}$^[4]

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${\displaystyle |\psi ⟩=|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},n_{{\mathbf{k}}_3}...n_{{\mathbf{k}}_l}...⟩}$ is obtained by applying a certain sum of permutation operators to the tensor product of eigenkets as follows:

^[7]: 16

This determinant is called theSlater determinant.^{[citation needed]} 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 identicalfermions must not occupy the same state (a statement of thePauli exclusion principle). Therefore, the occupation number of any single state is either 0 or 1. The eigenvalue associated to the fermionic Fock state${\displaystyle \hat{N_{{\mathbf{k}}_l}}}$ must be either 0 or 1.

Antisymmetric behaviour of Fermionic states under Exchange operator is taken care of by 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${\displaystyle |\psi ⟩=|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},...n_{{\mathbf{k}}_m}...n_{{\mathbf{k}}_l}...⟩}$ and want to shift a particle from state${\displaystyle k_l}$ to state${\displaystyle k_m}$, then we operate the Fock state by${\displaystyle c_{{\mathbf{k}}_m}^{†}.c_{{\mathbf{k}}_l}}$ in the following way:

Using the anticommutation relation we have

${\displaystyle c_{{\mathbf{k}}_m}^{†}.c_{{\mathbf{k}}_l}=-c_{{\mathbf{k}}_l}.c_{{\mathbf{k}}_m}^{†}}$

${\displaystyle c_{{\mathbf{k}}_m}^{†}.c_{{\mathbf{k}}_l}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},....n_{{\mathbf{k}}_m}...n_{{\mathbf{k}}_l}...⟩=\sqrt{n_{{\mathbf{k}}_m}+1}\sqrt{n_{{\mathbf{k}}_l}}|n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},....n_{{\mathbf{k}}_m}+\mathrm{1...}{n}_{{\mathbf{k}}_l}-\mathrm{1...}⟩}$

but,

Thus, fermionic Fock states are antisymmetric under operation by particle exchange operators.

Insecond quantization theory, theHamiltonian density function is given by

${\displaystyle \mathfrak{H}=\frac{1}{2m}{\mathrm{\nabla }}_i{\psi }^{\ast }(x){\mathrm{\nabla }}_i\psi (x)}$^[3]: 189

The totalHamiltonian is given by

In free Schrödinger theory,^[3]: 189

${\displaystyle \mathfrak{H}{\psi }_n^{(+)}(x)=-\frac{{\mathrm{\nabla }}^2}{2m}{\psi }_n^{(+)}(x)=E_n^0{\psi }_n^{(+)}(x)}$

and

${\displaystyle \int d^{3}x{\psi }_n^{(+{)}^{\ast }}(x){\psi }_{n^{\prime }}^{(+)}(x)={\delta }_{n{n}^{\prime }}}$

and

${\displaystyle \psi (x)=\sum _n{a}_n{\psi }_n^{(+)}(x)}$,

where${\displaystyle a_n}$ is the annihilation operator.

${\displaystyle \therefore \mathcal{H}=\sum _{n,n^{\prime }}\int d^{3}x{a}_{n^{\prime }}^{†}{\psi }_{n^{\prime }}^{(+{)}^{\ast }}(x)\mathfrak{H}{a}_n{\psi }_n^{(+)}(x)}$

Only for non-interacting particles do${\displaystyle \mathfrak{H}}$ and${\displaystyle a_n}$ commute; in general they do not commute. For non-interacting particles,

${\displaystyle \mathcal{H}=\sum _{n,n^{\prime }}\int d^{3}x{a}_{n^{\prime }}^{†}{\psi }_{n^{\prime }}^{(+{)}^{\ast }}(x)E_n^0{\psi }_n^{(+)}(x)a_n=\sum _{n,n^{\prime }}{E}_n^0{a}_{n^{\prime }}^{†}{a}_n{\delta }_{n{n}^{\prime }}=\sum _n{E}_n^0{a}_n^{†}{a}_n=\sum _n{E}_n^0\hat{N}}$

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.

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Thevacuum state or${\displaystyle |0⟩}$ is the state of the lowest energy and the expectation values of${\displaystyle a}$ and${\displaystyle a^{†}}$ vanish in this state:

${\displaystyle ⟨0|a|0⟩=⟨0|a^{†}|0⟩=0}$

The electric and magnetic fields and thevector potential have the mode expansion of the same general form:

${\displaystyle F\left(\overrightarrow{r},t\right)=\epsilon a{e}^{i\overrightarrow{k}\cdot \overrightarrow{r}-\omega t}+\epsilon a^{†}{e}^{-i\overrightarrow{k}\cdot \overrightarrow{r}-\omega t}}$

The expectation values of these field operators vanish in the vacuum state:

${\displaystyle ⟨0|F|0⟩=0}$

However, the expectation values of the square of these field operators are non-zero: there are field fluctuations in the vacuum state. Thesevacuum fluctuations are responsible for many interesting phenomena including theLamb shift inquantum optics.

In a multi-mode field each creation and annihilation operator operates on its own mode. So${\displaystyle a_{{\mathbf{k}}_l}}$ and${\displaystyle a_{{\mathbf{k}}_l}^{†}}$ will operate only on${\displaystyle |n_{{\mathbf{k}}_l}⟩}$. Since operators corresponding to different modes operate in different sub-spaces of the Hilbert space, the entire field is a direct product of${\displaystyle |n_{{\mathbf{k}}_l}⟩}$ over all the modes:

${\displaystyle |n_{{\mathbf{k}}_1}⟩|n_{{\mathbf{k}}_2}⟩|n_{{\mathbf{k}}_3}⟩\dots \equiv |n_{{\mathbf{k}}_1},n_{{\mathbf{k}}_2},n_{{\mathbf{k}}_3}...n_{{\mathbf{k}}_l}...⟩\equiv |\{n_{\mathbf{k}}\}⟩}$

The creation and annihilation operators operate on the multi-mode state by only raising or lowering the number state of their own mode:

We also define the totalnumber operator for the field which is a sum of number operators of each mode:

${\displaystyle {\hat{n}}_{\mathbf{k}}=\sum {\hat{n}}_{{\mathbf{k}}_l}}$

The multi-mode Fock state is an eigenvector of the total number operator whose eigenvalue is the total occupation number of all the modes

${\displaystyle {\hat{n}}_{\mathbf{k}}|\{n_{\mathbf{k}}\}⟩=\left(\sum n_{{\mathbf{k}}_l}\right)|\{n_{\mathbf{k}}\}⟩}$

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

${\displaystyle \hat{H}|\{n_{\mathbf{k}}\}⟩=\left(\sum \hslash \omega \left(n_{{\mathbf{k}}_l}+\frac{1}{2}\right)\right)|\{n_{\mathbf{k}}\}⟩}$

Single photons are routinely generated using single emitters (atoms, ions, molecules,Nitrogen-vacancy center,^[8]Quantum dot^[9]). However, these sources are not always very efficient, often presenting a low probability of actually getting a singlephoton 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 non-linearity of some materials like periodically poledLithium niobate (Spontaneous parametric down-conversion), or silicon (spontaneousFour-wave mixing) for example.

TheGlauber–Sudarshan P-representation of Fock states shows that these states are purely quantum mechanical and have no classical counterpart. The${\displaystyle \phi (\alpha )}$^{[clarification needed]} of these states in the representation is a${\displaystyle 2n}$'th derivative of theDirac delta function and therefore not a classicalprobability distribution.

• Coherent states
• Heisenberg limit
• Nonclassical light

• Vladan Vuletic ofMIT hasused an ensemble of atoms to produce a Fock state (a.k.a. single photon) sourceArchived 2023-06-15 at theWayback Machine (PDF)
• Produce and measure a single photon state (Fock state) with an interactive experimentQuantumLab

• Quantum optics
• Quantum field theory

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