Displacement operator

The displacement operator for one mode in quantum optics is the operator

${\hat {D}}(\alpha )=\exp \left(\alpha {\hat {a}}^{\dagger }-\alpha ^{\ast }{\hat {a}}\right)$,

where $\alpha$ is the amount of displacement in phase space, $\alpha ^{\ast }$ is the complex cojugate of that displacement, and ${\hat {a}}$ and ${\hat {a}}^{\dagger }$ are the lowering and raising operators, respectively. The effect of applying this operator in a similarity transformation of the ladder operators results in their displacement. Displaced states are eigenfunctions of the annihilation operator

Properties

The displacement operator is a unitary operator, and therefore obeys ${\hat {D}}(\alpha ){\hat {D}}^{\dagger }(\alpha )={\hat {D}}^{\dagger }(\alpha ){\hat {D}}(\alpha )=I$, where I is the identity matrix.

The effect of applying this operator in a similarity transformation of the ladder operators results in their displacement. Displaced states are eigenfunctions of the annihilation operator

Using the Baker-Campbell-Hausdorff formula#The Hadamard lemma, The effect of applying this operator in a similarity transformation of the ladder operators results in their displacement. Displaced states are eigenfunctions of the annihilation operator.
${\hat {D}}^{\dagger }(\alpha ){\hat {a}}{\hat {D}}(\alpha )={\hat {a}}+\alpha$
${\hat {D}}(\alpha ){\hat {a}}{\hat {D}}^{\dagger }(\alpha )={\hat {a}}-\alpha$

${\hat {D}}(\alpha ){\hat {D}}(\beta )=e^{i{\textrm {Im}}\left(\alpha \beta ^{\ast }\right)}{\hat {D}}(\alpha +\beta )$

When acting on an eigenket, the phase factor $e^{i{\textrm {Im}}\left(\alpha \beta ^{\ast }\right)}$ appears in each term of the resulting state, which makes it physically irrelevant.

Multimode displacement

The displacement operator can also be generalized to multimode displacement.

References

1. Gerry, Christopher, and Peter Knight. Introductory Quantum Optics.Cambridge (England): Cambridge UP, 2005.