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, ${\displaystyle {\hat {D}}(\alpha )|0\rangle =|\alpha \rangle }$ where ${\displaystyle |\alpha \rangle }$ is a coherent state. Displaced states are eigenfunctions of the annihilation (lowering) operator.

Alternative expressions

Two alternative ways to express the displacement operator are:

${\displaystyle {\hat {D}}(\alpha )=e^{-{\frac {1}{2}}|\alpha |^{2}}e^{+\alpha {\hat {a}}^{\dagger }}e^{-\alpha ^{*}{\hat {a}}}}$ 

${\displaystyle {\hat {D}}(\alpha )=e^{+{\frac {1}{2}}|\alpha |^{2}}e^{-\alpha ^{*}{\hat {a}}}e^{+\alpha {\hat {a}}^{\dagger }}}$ 

Multimode displacement

The displacement operator can also be generalized to multimode displacement.

References

Notes

See also

• Optical Phase Space