en>KylieTastic: Reverted edits by 49.244.170.121 (talk): editing tests (HG)

2014-12-22T17:15:28Z

Reverted edits by 49.244.170.121 (talk): editing tests (HG)

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	A '''Fock state''' (also known as a number state), in [[quantum mechanics]], is any element of a [[Fock space]] with a well-defined number of [[Elementary particle\|particle]]s (or [[quantum\|quanta]]). These states are named after the [[Soviet Union\|Soviet]] physicist, [[V. A. Fock]].		There is nothing to tell about myself at all.<br>Hurrey Im here and a part of wmflabs.org.<br>I really wish I am useful at all<br><br>my webpage - [https://www.youtube.com/user/EbonTalifarro Ebon Talifarro]

	~~==Definition==~~
	~~A more mathematical definition~~ is that Fock states are those elements of a Fock space which are eigenstates of the [[particle number operator]]. Elements of a Fock space which are superpositions of states of differing [[particle number]] (and thus not eigenstates of the number operator) are, therefore, not Fock states. Thus, not all elements of a Fock space are referred to ~~as "Fock states."~~

	If we limit to a single mode for simplicity (doing so we formally describe a mere [[harmonic oscillator]]), a Fock state is of the type <math>\|n\rangle</math> with ''n'' an integer value. This means that there are ''n'' quanta of excitation in the mode. <math>\|0\rangle</math> corresponds to the ground state (no excitation). It is different from '''0''', which is the null vector.

	~~Fock states form the most convenient [[basis (linear algebra)\|basis]] of the Fock space. They are defined to obey the following relations in the [[boson]]ic algebra:~~

	~~:<math>a^{\dagger}\|n\rangle=\sqrt{n+1}\|n+1\rangle</math>~~
	~~:<math>a\|n\rangle=\sqrt{n}\|n-1\rangle</math>~~
	~~:<math>\|n\rangle={1\over\sqrt{n!}}(a^{\dagger})^n\|0\rangle</math>~~

	~~with <math>a</math> (resp. <math>a^{\dagger}</math>) the [[Creation and annihilation operators\|annihilation (resp. creation) bose operator]]. Similar relations hold for [[fermion]]ic algebra.~~

	This allows to check that <math> a^{\dagger} a = n</math> and <math>{\rm Var}(a^{\dagger}a)=0</math>, i.e., that measuring the number of particles <math>a^{\dagger}a</math> in a Fock state always returns a definite value with no fluctuation.

	~~==Energy eigenstates==~~
	~~Fock states are [[eigenstates]] of the [[Hamiltonian (quantum mechanics)\|Hamiltonian]] of the field:~~

	~~:<math>H\|n\rangle=E_n\|n\rangle</math>~~

	~~where <math>E_n</math> is the energy [[eigenvalue]] corresponding to <math>\|n\rangle</math>. When we put in the expression for the Hamiltonian we get:~~

	~~:<math>\hbar \omega\left(a^{\dagger}a + \frac{1}{2} \right)\|n\rangle=\hbar \omega\left(n+\frac{1}{2}\right)\|n\rangle</math>~~

	~~Therefore energy of the state <math>\|n\rangle</math> is given by <math>\hbar \omega\left(n+\frac{1}{2}\right)</math> where <math>\omega</math> is the frequency of the field. Note that even~~ at ~~<math>n=0</math> the energy does not vanish. This is the [[zero-point energy]].~~

	~~==Vacuum fluctuations==~~
	~~The vacuum state or <math>\|0\rangle</math> is the state of lowest energy and the expectation values of <math>a</math> and <math>a^{\dagger}</math> vanish in this state:~~

	~~:<math>a\|0\rangle = 0 = \langle0\|a^{\dagger}</math>~~

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

	~~:<math>F(\vec{r},t) = \varepsilon a e^{i\vec{k}x-\omega t} + h.c </math>~~

	~~Thus it is easy to see that the expectation values of these field operators vanishes in the vacuum state:~~

	~~:<math>\langle0\|F\|0\rangle = 0 </math>~~

	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 quantum optics.

	~~==Multi-mode Fock states==~~
	In a multi-mode field each creation and annihilation operator operates on its own mode. So <math>a_{{\mathbf{k}}_{l}}</math> and <math>a^{\dagger}_{{\mathbf{k}}_{l}}</math> will operate only on <math>\|n_{{\mathbf{k}}_{l}}\rangle</math>. Since operators corresponding to different modes operate in different sub-spaces of the [[Hilbert space]], the entire field is a direct product of <math>\|n_{{\mathbf{k}}_l}\rangle </math> over all ~~the modes:~~

	~~:<math>\|n_{{\mathbf{k}}_{1}}\rangle \|n_{{\mathbf{k}}_{2}}\rangle \|n_{{\mathbf{k}}_{3}}\rangle... \equiv \|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}~~.~~..\rangle \equiv \|\{n_{\mathbf{k}}\}\rangle~~ <~~/math~~>

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

	:<math> a_{{\mathbf{k}}_l} \|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}},...\rangle = \sqrt{n_{{\mathbf{k}}_{l}}} \|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}-1 ,...\rangle </math>

	~~:<math>~~ a^{\dagger}_{{\mathbf{k}}_l} \|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}},...\rangle = \sqrt{n_{{\mathbf{k}}_{l}} +1 } \|n_{{\mathbf{k}}_{1}}, n_{{\mathbf{k}}_{2}} ,n_{{\mathbf{k}}_{3}}...n_{{\mathbf{k}}_{l}}+1 ,...\rangle </math>

	~~We also define the total [[number operator]] for the field which is a sum of number operators of each mode:~~
	~~:<math> \hat{n}_{\mathbf{k}} = \sum \hat{n}_{\mathbf{k}_l} </math>~~

	~~The multi-mode Fock state is an eigenvector of the total number operator whose eigenvalue is the total occupation number of all the modes~~
	~~:<math> \hat{n}_{\mathbf{k}} \|\{n_{\mathbf{k}}\}\rangle = \bigg( \sum n_{\mathbf{k}_l} \bigg) \|\{n_{\mathbf{k}}\}\rangle </math>~~

	~~The multi-mode Fock states are also eigenstates~~ of ~~the multi-mode Hamiltonian~~

	~~:<math> \hat{H} \|\{n_{\mathbf{k}}\}\rangle = \bigg( \sum \hbar \omega \big(n_{\mathbf{k}_l} + \frac{1}{2} \big)\bigg) \|\{n_{\mathbf{k}}\}\rangle </math>~~

	~~==Source of single photon state==~~

	~~Single photons are routinely generated using single emitters (atoms, [[Nitrogen-vacancy center]]~~
	~~,<ref>C. Kurtsiefer, S. Mayer, P. Zarda, Patrick and H. Weinfurter, (2000), "Stable Solid-State Source of Single Photons",~~
	~~''Phys. Rev. Lett.'' '''85''' (2) 290--293, [http://link.aps~~.org~~/doi/10.1103/PhysRevLett.85.290 doi 10~~.~~1103/PhysRevLett.85.290]~~<~~/ref~~> ~~[[Quantum dot]]~~ <~~ref~~>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]
	<~~/ref~~>~~). However, these sources are not always very efficient (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 an 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]] ([[Spontaneous parametric down-conversion]]), or silicon (spontaneous [[Four-wave mixing]]) 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 <math>\scriptstyle\varphi(\alpha) \,</math> of these states in the representation is a <math>2n</math>'th derivative of the [[Dirac delta function]] and therefore not a classical probability distribution.

	~~==See also==~~
	*[[Coherent states]]
	*[[Heisenberg limit]]
	*[~~[Nonclassical light]]~~

	~~==References==~~
	~~{{reflist\|2}}~~

	~~==External links==~~
	*Vladan Vuletic of [[MIT]] has [http://www.~~rle.mit~~.~~edu/eap~~/~~other~~/~~single_photon_source_article.pdf 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 [http://www.QuantumLab.de QuantumLab]

	~~[[Category:Quantum mechanics]]~~
	~~[[Category:Quantum optics]]~~
	~~[[Category:Quantum field theory]~~]

en>Toby Bartels: /* Proof using the hyperreals */ Missing asterisks. Consistent formatting of "st".

2013-12-16T20:19:37Z

Proof using the hyperreals: Missing asterisks. Consistent formatting of "st".

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en>ChrisGualtieri: /* History */Typo fixing, typos fixed: the 1830's → the 1830s using AWB

2012-09-01T02:47:45Z

History: Typo fixing, typos fixed: the 1830's → the 1830s using AWB

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Extreme value theorem - Revision history

en>KylieTastic: Reverted edits by 49.244.170.121 (talk): editing tests (HG)

en>Toby Bartels: /* Proof using the hyperreals */ Missing asterisks. Consistent formatting of "st".

en>ChrisGualtieri: /* History */Typo fixing, typos fixed: the 1830's → the 1830s using AWB