In the quantum mechanics study of optical phase space, the displacement operator for one mode is the shift operator in quantum optics, D ^ = exp ⁡ {\displaystyle {\hat {D}}=\exp \left},where α {\displaystyle \alpha } is the amount of displacement in optical phase space, α ∗ {\displaystyle \alpha ^{*}} is the complex conjugate of that displacement, and a ^ {\displaystyle {\hat {a}}} and a ^ † {\displaystyle {\hat {a}}^{\dagger }} are the lowering and raising operators, respectively. Wikipedia