Displacement operator: Difference between revisions

Revision as of 07:13, 16 January 2012

Revision as of 14:55, 31 January 2012

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

where <math>\alpha</math>''α'' is the amount of displacement in [[optical phase space]], ''α''<mathsup>\alpha^\ast*</mathsup> is the complex conjugate of that displacement, and <math>\hat{a} </math>''â'' and ''â''<mathsup>\hat{a}^\dagger†</mathsup> 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 <math>\alpha</math>''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>''α''⟩ is a [[coherent state]].

Displaced states are [[eigenfunctions]] of the annihilation (lowering) operator.