Fock state

 ( Quantum Optics ) 

where $\{\backslash \{\; \backslash \; ,\; \backslash \; \backslash \}\; \}$ is the commutator and $\backslash delta\_\{i\; j\}$ is the Kronecker delta. These anticommutation relations can be used to show antisymmetric behaviour of Fermionic Fock states.

$\backslash widehat\{N\_\_l\}\}|n\_\_\{1\}\},\; n\_\_\{2\}\}\; ,n\_\_\{3\}\}...n\_\_\{l\}\}...\backslash rangle=n\_\_\{l\}\}\; |n\_\_\{1\}\},\; n\_\_\{2\}\}\; ,n\_\_\{3\}\}...n\_\_\{l\}\}...\backslash rangle$

 \left|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3} ... n_{\mathbf{k}_l} ...\right\rangle =
 S_-\left|i_1, i_2, i_3 ... i_l ...\right\rangle =
 \frac{1}{\sqrt{N!}}\begin{vmatrix}
   \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
     

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 must not occupy the same state (a statement of the Pauli exclusion principle). Therefore, the occupation number of any single state is either 0 or 1. The eigenvalue associated to the fermionic Fock state $\backslash widehat\{N\_\_l\}\}$ must be either 0 or 1.

Using the anticommutation relation we have

$c\_\{\backslash mathbf\{k\}\_m\}^\backslash dagger.c\_\{\backslash mathbf\{k\}\_l\}\; =\; -c\_\{\backslash mathbf\{k\}\_l\}.c\_\{\backslash mathbf\{k\}\_m\}^\backslash dagger$

$$

 c_{\mathbf{k}_m}^{\dagger}.c_{\mathbf{k}_l} \left|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, .... n_{\mathbf{k}_m} ... n_{\mathbf{k}_l} ... \right\rangle =
 \sqrt{n_{\mathbf{k}_m} + 1}\sqrt{n_{\mathbf{k}_l}} \left|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, .... n_{\mathbf{k}_m} + 1 ... n_{\mathbf{k}_l} - 1 ...\right\rangle
     

but, $$\backslash begin\{align\}$$

      &c__{l}}.c__{m}}^{\dagger}|n__{1}}, n__{2}}, ....n__{m}}... n__{l}}...\rangle \\
 ={} -&c__{m}}^{\dagger}.c__{l}}|n__{1}}, n__{2}}, .... n__{m}}... n__{l}}...\rangle \\
 ={} -&\sqrt{n__{m}} +1 }\sqrt{n__{l}}}|n__{1}}, n__{2}}, .... n__{m}} + 1 ... n__{l}} - 1...\rangle
     

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

$\backslash mathfrak\{H\}\; =\; \backslash frac\{1\}\{2m\}\; \backslash nabla\_\{i\}\backslash psi^\{*\}(x)\backslash ,\; \backslash nabla\_\{i\}\backslash psi(x)$

The total Hamiltonian is given by

$\backslash begin\{align\}$

             \mathcal{H} &= \int d^3 x\,\mathfrak{H}
                          = \int d^{3}x \psi^{*}(x)\left(-\frac{\nabla^2}{2m}\right)\psi(x) \\
 \therefore \mathfrak{H} &= -\frac{\nabla^2}{2m}
     

In free Schrödinger theory,

$\backslash mathfrak\{H\}\backslash psi\_\{n\}^\{(+)\}(x)\; =\; -\backslash frac\{\backslash nabla^2\}\{2m\}\backslash psi\_\{n\}^\{(+)\}(x)\; =\; E\_\{n\}^\{0\}\backslash psi\_\{n\}^\{(+)\}(x)$

and

$\backslash int\; d^3\; x\backslash ,\; \backslash psi\_\{n\}^\{(+)^\{*\}\}(x)\backslash ,\; \backslash psi\_\{n\text{'}\}^\{(+)\}(x)\; =\; \backslash delta\_\{nn\text{'}\}$

and

$\backslash psi(x)\; =\; \backslash sum\_n\; a\_n\; \backslash psi\_\{n\}^\{(+)\}(x)$,

where $a\_n$ is the annihilation operator.

$\backslash therefore\; \backslash mathcal\{H\}\; =\; \backslash sum\_\{n,n\text{'}\}\backslash int\; d^\{3\}x\backslash ,\; a^\{\backslash dagger\}\_\{n\text{'}\}\backslash psi\_\{n\text{'}\}^\{(+)^\{*\}\}(x)\backslash ,\; \backslash mathfrak\{H\}a\_n\; \backslash psi\_\{n\}^\{(+)\}(x)$

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

$\backslash mathcal\{H\}\; =$

 \sum_{n,n'}\int d^3 x\, a^{\dagger}_{n'}\psi_{n'}^{(+)^{*}}(x)\, E^{0}_{n}\psi_{n}^{(+)}(x)a_n =
 \sum_{n,n'}E^{0}_{n} a^{\dagger}_{n'}a_n\delta_{nn'} =
 \sum_{n}E^{0}_{n}a^{\dagger}_n a_n =
 \sum_{n}E^{0}_{n}\widehat{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.

$\backslash langle\; 0\; |\; a|0\backslash rangle\; =\; \backslash langle\; 0\; |a^\backslash dagger|0\backslash rangle\; =\; 0$

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

$F\backslash left(\backslash vec\{r\},\; t\backslash right)\; =\; \backslash varepsilon\; a\; e^\{i\backslash vec\{k\}\backslash cdot\backslash vec\{r\}\; -\; \backslash omega\; t\}\; +\; \backslash varepsilon\; a^\backslash dagger\; e^\{-i\backslash vec\{k\}\backslash cdot\backslash vec\{r\}\; -\; \backslash omega\; t\}$

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

$\backslash langle\; 0|F|0\; \backslash rangle\; =\; 0$

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.

 \left|n_{\mathbf{k}_1}\right\rangle \left|n_{\mathbf{k}_2}\right\rangle \left|n_{\mathbf{k}_3}\right\rangle \ldots \equiv
 \left|n_{\mathbf{k}_1}, n_{\mathbf{k}_2}, n_{\mathbf{k}_3}... n_{\mathbf{k}_l}... \right\rangle \equiv
 \left|\{n_\mathbf{k}\}\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:

$\backslash begin\{align\}$

 a__l} |n__{1}}, n__{2}}, n__{3}}... n__{l}}, ...\rangle &=
 \sqrt{n__{l}}} |n__{1}}, n__{2}}, n__{3}}... n__{l}}-1, ...\rangle \\
 a^{\dagger}__l} |n__{1}}, n__{2}}, n__{3}}... n__{l}},...\rangle &=
 \sqrt{n__{l}} +1 } |n__{1}}, n__{2}}, n__{3}}... n__{l}} + 1, ...\rangle
     

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

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

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

$\backslash hat\{n\}\_\{\backslash mathbf\{k\}\}\; |\backslash \{n\_\{\backslash mathbf\{k\}\}\backslash \}\backslash rangle\; =\; \backslash left(\; \backslash sum\; n\_\{\backslash mathbf\{k\}\_l\}\; \backslash right)\; |\backslash \{n\_\{\backslash mathbf\{k\}\}\backslash \}\backslash rangle$

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

$\backslash hat\{H\}\; \backslash left|\backslash \{n\_\{\backslash mathbf\{k\}\}\backslash \}\backslash right\backslash rangle\; =\; \backslash left(\; \backslash sum\; \backslash hbar\; \backslash omega\; \backslash left(n\_\{\backslash mathbf\{k\}\_l\}\; +\; \backslash frac\{1\}\{2\}\; \backslash right)\backslash right)\; \backslash left|\backslash \{n\_\{\backslash mathbf\{k\}\}\backslash \}\backslash right\backslash rangle$

 ''Phys. Rev. Lett.'' '''85''' (2) 290--293, [http://link.aps.org/doi/10.1103/PhysRevLett.85.290    doi 10.1103/PhysRevLett.85.290] [[Quantum dot]]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--1505 [http://link.aps.org/doi/10.1103/PhysRevLett.86.1502 DOI 10.1103/PhysRevLett.86.1502]
     

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.

• Coherent states
• Heisenberg limit
• Nonclassical light

• Vladan Vuletic of MIT has 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 experiment QuantumLab