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Cuzkatzimhut (talk | contribs) m formatting |
→Multimode displacement: added a definition of mutimode displacement operator |
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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\textbf{k}\psi(\textbf{k})\hat a(\textbf{k})^{\dagger}</math>,
where <math>\textbf{k}</math> is the wave vector and its magnitude is related to the frequency <math>\omega_{\textbf{k}}</math> according to <math>|\textbf{k}|=\omega_{\textbf{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==
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