Displacement operator: Difference between revisions

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{{Quantum optics operators}}

The '''displacement operator''' for one mode in [[quantum optics]] is the [[operator]]

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

where <math>\alpha</math> is the amount of displacement in [[optical phase space]], <math>\alpha^\ast</math> is the complex conjugate of that displacement, and <math>\hat{a} </math> and <math>\hat{a}^\dagger</math> are the [[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>. 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.

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

== Properties ==

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. Specifically, this can be done by utilizing the [[Baker-Campbell-Hausdorff formula#The Hadamard lemma|Baker-Campbell-Hausdorff formula]].

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