Displacement operator: Difference between revisions

Revision as of 04:36, 13 November 2008

Revision as of 01:39, 19 May 2009

Line 9:

The name of this operator is derived from its ability to displace a localized state in phase space by a magnitude <math>\alpha</math>. 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>.

Displaced states are eigenfunctions of the annihilation (lowering) operator.<br>

== Properties ==

Line 19:

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

<math>\hat{D}(\alpha)\hat{D}(\beta)=e^{\mathrm i\cdot\operatorname{Im} \left(\alpha \beta^\ast \right)}\hat{D}(\alpha + \beta)</math><br>

When acting on an eigenket, the phase factor <math>e^{\mathrm i\cdot\operatorname{Im} \left(\alpha \beta^\ast \right)}</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>

Line 30:

==Notes==

==See Alsoalso==

* [[Optical Phase Space]]

[[Category:Quantum optics]]

{{math-stub}}

{{physics-stub}}

==See Also==

* [[Optical Phase Space]]