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| {{Context|date=October 2009}}
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| The '''displacement operator''' for one mode in [[quantum optics]] is the [[Operator (mathematics)|operator]]
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| :<math>\hat{D}(\alpha)=\exp \left ( \alpha \hat{a}^\dagger - \alpha^\ast \hat{a} \right ) </math>, | |
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| 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.
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| 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]].
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| Displaced states are [[eigenfunctions]] of the annihilation (lowering) operator.
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| == Properties == | |
| The displacement operator is a [[unitary operator]], and therefore obeys
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| <math>\hat{D}(\alpha)\hat{D}^\dagger(\alpha)=\hat{D}^\dagger(\alpha)\hat{D}(\alpha)=I</math>,
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| 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.
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| :<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>
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| 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]].
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| :<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>
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| which shows us that:
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| :<math>\hat{D}(\alpha)\hat{D}(\beta)= e^{(\alpha\beta^*-\alpha^*\beta)/2} \hat{D}(\alpha + \beta)</math>
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| 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>
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| == Alternative expressions ==
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| Two alternative ways to express the displacement operator are:
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| :<math>\hat{D}(\alpha) = e^{ -\frac{1}{2} | \alpha |^2 } e^{+\alpha \hat{a}^{\dagger}} e^{-\alpha^{*} \hat{a} } </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>
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| == Multimode displacement ==
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| The displacement operator can also be generalized to multimode displacement. A multimode creation operator can be defined as
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| :<math>\hat A_{\psi}^{\dagger}=\int d\mathbf{k}\psi(\mathbf{k})\hat a(\mathbf{k})^{\dagger}</math>,
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| 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
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| :<math>\hat{D}_{\psi}(\alpha)=\exp \left ( \alpha \hat A_{\psi}^{\dagger} - \alpha^\ast \hat A_{\psi} \right ) </math>,
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| and define the multimode coherent state as
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| :<math>|\alpha_{\psi}\rangle\equiv\hat{D}_{\psi}(\alpha)|0\rangle</math>.
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| ==References== | |
| <references />
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| ==Notes==
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| {{Empty section|date=July 2010}}
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| ==See also==
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| * [[Optical Phase Space]]
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| {{Physics operators}}
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| [[Category:Quantum optics]]
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