Displacement Operator

The unitary shifting or displacement operator

$D_{\alpha}=e^{\alpha a^{\dagger}-\alpha^* a}$

causes a shift of the operator $a$

$D^{\dagger}_{\alpha}aD_{\alpha}=e^{-\alpha a^{\dagger}+\alpha^* a}ae^{\alpha a^{\dagger}-\alpha^* a}=a+\alpha$

for an arbitrary complex number $α$ .

The displacement operator has the following properties

$D^{\dagger}(\alpha)=D^{-1}(\alpha)=D(-\alpha)$

$D^{\dagger}(\alpha)aD(\alpha)=a+\alpha$

$D^{\dagger}(\alpha)a^{\dagger}D(\alpha)=a^{\dagger}+\alpha^*$

From "Quantum Optics" by Walls and Milburn, Springer (1994), pp. 12