Displacement operator

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

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

where ${\displaystyle \alpha }$ is the amount of displacement in optical phase space, ${\displaystyle \alpha ^{\ast }}$ is the complex conjugate of that displacement, and ${\displaystyle {\hat {a}}}$ and ${\displaystyle {\hat {a}}^{\dagger }}$ 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 ${\displaystyle \alpha }$. It may also act on the vacuum state by displacing it into a coherent state. Specifically, ${\displaystyle {\hat {D}}(\alpha )|0\rangle =|\alpha \rangle }$. Displaced states are eigenfunctions of the annihilation (lowering) operator.

Properties

The displacement operator is a unitary operator, and therefore obeys ${\displaystyle {\hat {D}}(\alpha ){\hat {D}}^{\dagger }(\alpha )={\hat {D}}^{\dagger }(\alpha ){\hat {D}}(\alpha )=I}$ , where I is the identity matrix. Since ${\displaystyle {\hat {D}}^{\dagger }(\alpha )={\hat {D}}(-\alpha )}$ , the hermitian conjugate of the displacement operator can also be interpreted as a displacement of opposite magnitude (${\displaystyle -\alpha }$ ). 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.

${\displaystyle {\hat {D}}^{\dagger }(\alpha ){\hat {a}}{\hat {D}}(\alpha )={\hat {a}}+\alpha }$ 
${\displaystyle {\hat {D}}(\alpha ){\hat {a}}{\hat {D}}^{\dagger }(\alpha )={\hat {a}}-\alpha }$ 

${\displaystyle {\hat {D}}(\alpha ){\hat {D}}(\beta )=e^{\mathrm {i} \cdot \operatorname {Im} \left(\alpha \beta ^{\ast }\right)}{\hat {D}}(\alpha +\beta )}$ 

When acting on an eigenket, the phase factor ${\displaystyle e^{\mathrm {i} \cdot \operatorname {Im} \left(\alpha \beta ^{\ast }\right)}}$  appears in each term of the resulting state, which makes it physically irrelevant.^[1]

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.

Notes

See also

• Optical Phase Space