Displacement Operator

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

,

where α is the amount of displacement in optical phase space, α* is the complex conjugate of that displacement, and â and ↠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 α. It may also act on the vacuum state by displacing it into a coherent state. Specifically, where |α> is a coherent state. Displaced states are eigenfunctions of the annihilation (lowering) operator.