Displacement operator: Difference between revisions

Revision as of 15:31, 2 March 2012

Revision as of 01:16, 26 June 2012

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{{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''^* is the complex conjugate of that displacement, and ''â'' and ''â''^† are the [[creation and annihilation operators|lowering and raising operators]], respectively.

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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 [[similarity transformation]] of the ladder operators results in their displacement.

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

:<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^{(\beta\alpha^*-\alpha\beta^*)/2}. </math>

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:<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>,

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==Notes==

{{Empty section|date=July 2010}}

==See also==

* [[Optical Phase Space]]

{{Physics operators}}

[[Category:Quantum optics]]

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