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Wikipedia / Fock state

In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particless (or quanta). These states are named after the Soviet physicist Vladimir Fock. Fock states play an important role in the second quantization formulation of quantum mechanics.

The particle representation was first treated in detail by Paul Dirac for bosons and by Pascual Jordan and Eugene Wigner for fermions. The Fock states of bosons and fermions obey useful relations with respect to the Fock space creation and annihilation operators.

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, the tensor products must be alternating (anti-symmetric) or symmetric products of the underlying one-particle Hilbert spaces. Specifically:

If the number of particles is variable, one constructs the Fock space as the direct sum of the tensor product Hilbert spaces for each particle 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.

be an orthonormal 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 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 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).

denotes the number of particles in the i-th state ki, acts on the Fock state in the following way:

.

Fock states often form the most convenient basis of a Fock space. Elements of a Fock space that are superpositions of states of differing particle 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".

as

the definition of Fock state ensures that the variance, i.e., measuring the number of particles in a Fock state always returns a definite value with no fluctuation.

Example using two particles

, and any operator, we have the following condition for indistinguishability:

  1. .

for bosons for fermions are arbitrary, we can say,

  1. for bosons and

  2. 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.

Bosonic Fock state

Bosons, which are particles with integer spin, follow a simple rule: their composite eigenstate is symmetric under operation by an exchange operator. For example, in a two particle system .

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, respectively. The action of these operators on a Fock state are given by the following two equations:

  • :

  • :

Non-Hermiticity of creation and annihilation operators

The bosonic Fock state creation and annihilation operators are not Hermitian operators.

Operator identities

The commutation relations of creation and annihilation operators in a bosonic system are

is the commutator is the Kronecker delta.

N bosonic basis states

Action on some specific Fock states

Action of number operators

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 in the following way:

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

Fermionic Fock state

Occupation number representation

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.

Fermion creation and annihilation operators

To retain the antisymmetric behaviour of fermions as:

  • acts on the basis states as:

  • acts as:

Operator identities

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

is the anticommutator is the Kronecker delta. These anticommutation relations can be used to show antisymmetric behaviour of Fermionic Fock states.

Action of number operators

for Fermions.

Maximum occupation number

is obtained by applying a certain sum of permutation operators to the tensor product of eigenkets as follows:

This determinant is called the Slater determinant. 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 fermions must not occupy the same state (a statement of the Pauli exclusion principle must be either 0 or 1.

Action on some specific Fock states

Antisymmetric behaviour of Fermionic Fock state

in the following way:

Using the anticommutation relation we have

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

Fock states are not energy eigenstates in general

In second quantization theory, the Hamiltonian density function is given by

The total Hamiltonian is given by

In free Schrödinger theory,

and

and

  1. ,

is the annihilation operator.

commute; in general they do not commute. For non-interacting particles,

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 vanish in this state:

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

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

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

Multi-mode Fock states

over all the modes:

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 total number operator for the field which is a sum of number operators of each mode:

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

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

Source of single photon state

Single photons are routinely generated using single emitters (atoms, ions, molecules, Nitrogen-vacancy center, Quantum dot). 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 non-linearity of some materials like periodically poled Lithium niobate (Spontaneous parametric down-conversion), or silicon (spontaneous Four-wave mixing) for example.

Non-classical behaviour

The Glauber–Sudarshan P-representation'th derivative of the Dirac delta function and therefore not a classical probability distribution.

See also

References