en>Yobot: WP:CHECKWIKI error fixes + other fixes using AWB (10065)

2014-03-28T20:44:15Z

WP:CHECKWIKI error fixes + other fixes using AWB (10065)

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	~~{{Context\|date=October 2009}}~~		Yoshiko is her title but she doesn't like when people use her full name. His working day job is a cashier and his salary has auto warranty been really [http://Www.Wikihow.com/Buy-an-Extended-Warranty-for-a-Car fulfilling]. The auto [http://www.dutchtaskforce.com/index.php?mod=users&action=view&id=46500 car warranty] preferred pastime for my kids and me is taking part in [http://www.dmv.org/buy-sell/auto-warranty/extended-warranty.php crochet] and now I'm trying to earn money with it. Her spouse and her [http://calvaryhill.net/xe/board_DsjO50/23803 extended auto warranty] chose to reside in Delaware but she needs to transfer simply because of her auto warranty family members.<br><br>My weblog: [http://www.team-qnak.com/qnak/index.php?mod=users&action=view&id=8348 www.team-qnak.com]

	~~The~~ '~~''displacement operator''' for one mode in [[quantum optics]]~~ is ~~the~~ [~~[Operator (mathematics)\|operator]]~~

	:~~<math>\hat{D}(\alpha)=\exp \left ( \alpha \hat{a}^\dagger - \alpha^\ast \hat{a} \right ) <~~/~~math>,~~

	~~where ''α'' is the amount of displacement in [[optical phase space]], ''α''<sup>*<~~/~~sup> is the complex conjugate of that displacement, and ''â'' and ''â''<sup>†<~~/~~sup> are the [[creation and annihilation operators\|lowering and raising operators]~~]~~, respectively~~.
	The ~~name of this operator is derived from its ability to displace a localized state in phase space by a magnitude ''α''. It may also act on the vacuum state by displacing it into a~~ [~~[coherent state]]. Specifically,~~
	~~<math>\hat{D}(\alpha)\|0\rangle=\|\alpha\rangle<~~/~~math> where <math>\|\alpha\rangle<~~/~~math> is a [[coherent state]]~~.
	~~Displaced states are [[eigenfunctions]] of the annihilation (lowering) operator~~.

	== ~~Properties~~ ==
	~~The displacement operator is a [[unitary operator~~]], and ~~therefore obeys~~
	~~<math>\hat{D}(\alpha)\hat{D}^\dagger(\alpha)=\hat{D}^\dagger(\alpha)\hat{D}(\alpha)=I</math>,~~
	~~where I~~ is the identity matrix. Since <math> \hat{D}^\dagger(\alpha)=\hat{D}(-\alpha)</math>, the [[hermitian conjugate]] of the displacement operator can also be interpreted as a displacement of opposite magnitude (<math>-\alpha</math>). The effect of applying this operator in a [~~[matrix similarity\|similarity transformation]] of the ladder operators results in their displacement.~~

	:~~<math>\hat{D}^\dagger(\alpha) \hat{a} \hat{D}(\alpha)=\hat{a}+\alpha<~~/~~math>~~
	~~:<math>\hat{D}(\alpha) \hat{a} \hat{D}^\dagger(\alpha)=\hat{a}-\alpha<~~/~~math>~~

	~~The product of two displacement operators is another displacement operator, apart from a phase factor, has the total displacement as the sum of the two individual displacements~~. ~~This can be seen by utilizing the [[Baker-Campbell-Hausdorff formula#The Hadamard lemma\|Baker-Campbell-Hausdorff formula]]~~.

	:<math> e^{\alpha \hat{a}^{\dagger} - \alpha^\hat{a}} e^{\beta\hat{a}^{\dagger} - \beta^\hat{a}} = e^{(\alpha + \beta)\hat{a}^{\dagger} - (\beta^+\alpha^)\hat{a}} e^{(\alpha\beta^-~~\alpha^\beta)~~/~~2}. </math>~~

	~~which shows us that:~~

	:<math>\hat{D}(\alpha)\hat{D}(\beta)= e^{(\alpha\beta^-~~\alpha^\beta)/2} \hat{D}(\alpha + \beta)<~~/~~math>~~

	When acting on an eigenket, the phase factor <math>e^{(\alpha\beta^-~~\alpha^\beta)/2}</math> appears in each term of the resulting state, which makes it physically irrelevant~~.~~<ref>Gerry, Christopher,~~ and ~~Peter Knight: ''Introductory Quantum Optics'~~'. ~~Cambridge (England)~~: ~~Cambridge UP, 2005~~.</~~ref>~~

	~~== Alternative expressions ==~~
	~~Two alternative ways to express the displacement operator are:~~
	~~:<math>\hat{D}(\alpha) = e^{ -\frac{1}{2} \| \alpha \|^2 } e^{+\alpha \hat{a}^{\dagger}} e^{-\alpha^{*} \hat{a} } <~~/~~math>~~

	~~:<math>\hat{D}(\alpha) = e^{ +\frac{1}{2} \| \alpha \|^2 } e^{-\alpha^{*} \hat{a} }e^{+\alpha \hat{a}^{\dagger}} <~~/~~math>~~

	~~== Multimode displacement ==~~
	~~The displacement operator can also be generalized~~ to ~~multimode displacement~~. ~~A multimode creation operator can be defined as~~

	:<~~math~~>~~\hat A_{\psi}^{\dagger}=\int d\mathbf{k}\psi(\mathbf{k})\hat a(\mathbf{k})^{\dagger}~~<~~/math~~>,

	~~where <math>\mathbf{k}</math> is the wave vector and its magnitude is related to the frequency <math>\omega_{\mathbf{k}}</math> according to <math>\|\mathbf{k}\|=\omega_{\mathbf{k}}~~/c</~~math>~~. ~~Using this definition, we can write the multimode displacement operator as~~

	~~:<math>\hat{D}_{\psi}(\alpha)=\exp \left ( \alpha \hat A_{\psi}^{\dagger}~~ - ~~\alpha^\ast \hat A_{\psi} \right ) <~~/~~math>,~~

	~~and define the multimode coherent state as~~

	~~:<math>\|\alpha_{\psi}\rangle\equiv\hat{D}_{\psi}(\alpha)\|0\rangle<~~/~~math>~~.

	==~~References~~==
	~~<references />~~

	~~==Notes==~~
	~~{{Empty section\|date=July 2010}}~~

	~~==See also==~~

	* [[Optical Phase Space]]

	~~{{Physics operators}}~~

	~~[[Category:Quantum optics]~~]

en>EagleFan: Repairing link to disambiguation page - You can help!

2010-05-28T11:22:10Z

Repairing link to disambiguation page - You can help!

New page

{{Context|date=October 2009}}

The '''displacement operator''' for one mode in [[quantum optics]] is the [[Operator (mathematics)|operator]]

:<math>\hat{D}(\alpha)=\exp \left ( \alpha \hat{a}^\dagger - \alpha^\ast \hat{a} \right ) </math>,

where ''α'' is the amount of displacement in [[optical phase space]], ''α''<sup>*</sup> is the complex conjugate of that displacement, and ''â'' and ''â''<sup>†</sup> are the [[creation and annihilation operators|lowering and raising operators]], respectively.
The name of this operator is derived from its ability to displace a localized state in phase space by a magnitude ''α''. It may also act on the vacuum state by displacing it into a [[coherent state]]. Specifically,
<math>\hat{D}(\alpha)|0\rangle=|\alpha\rangle</math> where <math>|\alpha\rangle</math> is a [[coherent state]].
Displaced states are [[eigenfunctions]] of the annihilation (lowering) operator.

== Properties ==
The displacement operator is a [[unitary operator]], and therefore obeys
<math>\hat{D}(\alpha)\hat{D}^\dagger(\alpha)=\hat{D}^\dagger(\alpha)\hat{D}(\alpha)=I</math>,
where I is the identity matrix. Since <math> \hat{D}^\dagger(\alpha)=\hat{D}(-\alpha)</math>, the [[hermitian conjugate]] of the displacement operator can also be interpreted as a displacement of opposite magnitude (<math>-\alpha</math>). The effect of applying this operator in a [[matrix similarity|similarity transformation]] of the ladder operators results in their displacement.

:<math>\hat{D}^\dagger(\alpha) \hat{a} \hat{D}(\alpha)=\hat{a}+\alpha</math>
:<math>\hat{D}(\alpha) \hat{a} \hat{D}^\dagger(\alpha)=\hat{a}-\alpha</math>

The product of two displacement operators is another displacement operator, apart from a phase factor, has the total displacement as the sum of the two individual displacements. This can be seen by utilizing the [[Baker-Campbell-Hausdorff formula#The Hadamard lemma|Baker-Campbell-Hausdorff formula]].

:<math> e^{\alpha \hat{a}^{\dagger} - \alpha^*\hat{a}} e^{\beta\hat{a}^{\dagger} - \beta^*\hat{a}} = e^{(\alpha + \beta)\hat{a}^{\dagger} - (\beta^*+\alpha^*)\hat{a}} e^{(\alpha\beta^*-\alpha^*\beta)/2}. </math>

which shows us that:

:<math>\hat{D}(\alpha)\hat{D}(\beta)= e^{(\alpha\beta^*-\alpha^*\beta)/2} \hat{D}(\alpha + \beta)</math>

When acting on an eigenket, the phase factor <math>e^{(\alpha\beta^*-\alpha^*\beta)/2}</math> appears in each term of the resulting state, which makes it physically irrelevant.<ref>Gerry, Christopher, and Peter Knight: ''Introductory Quantum Optics''. Cambridge (England): Cambridge UP, 2005.</ref>

== Alternative expressions ==
Two alternative ways to express the displacement operator are:
:<math>\hat{D}(\alpha) = e^{ -\frac{1}{2} | \alpha |^2 } e^{+\alpha \hat{a}^{\dagger}} e^{-\alpha^{*} \hat{a} } </math>

:<math>\hat{D}(\alpha) = e^{ +\frac{1}{2} | \alpha |^2 } e^{-\alpha^{*} \hat{a} }e^{+\alpha \hat{a}^{\dagger}} </math>

== Multimode displacement ==
The displacement operator can also be generalized to multimode displacement. A multimode creation operator can be defined as

:<math>\hat A_{\psi}^{\dagger}=\int d\mathbf{k}\psi(\mathbf{k})\hat a(\mathbf{k})^{\dagger}</math>,

where <math>\mathbf{k}</math> is the wave vector and its magnitude is related to the frequency <math>\omega_{\mathbf{k}}</math> according to <math>|\mathbf{k}|=\omega_{\mathbf{k}}/c</math>. Using this definition, we can write the multimode displacement operator as

:<math>\hat{D}_{\psi}(\alpha)=\exp \left ( \alpha \hat A_{\psi}^{\dagger} - \alpha^\ast \hat A_{\psi} \right ) </math>,

and define the multimode coherent state as

:<math>|\alpha_{\psi}\rangle\equiv\hat{D}_{\psi}(\alpha)|0\rangle</math>.

==References==
<references />

==Notes==
{{Empty section|date=July 2010}}

==See also==

* [[Optical Phase Space]]

{{Physics operators}}

[[Category:Quantum optics]]

Constraint algebra - Revision history

en>Yobot: WP:CHECKWIKI error fixes + other fixes using AWB (10065)

en>EagleFan: Repairing link to disambiguation page - You can help!