The displacement operator for one mode in quantum optics is the shift operator
D ^ ( α ) = exp ( α a ^ † − α ∗ a ^ ) ,
where α is the amount of displacement in optical phase space, α ∗ is the complex conjugate of that displacement, and a ^ and a ^ † 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, D ^ ( α ) | 0 ⟩ = | α ⟩ where | α ⟩ is a coherent state, which is the eigenstates of the annihilation (lowering) operator.
The displacement operator is a unitary operator, and therefore obeys D ^ ( α ) D ^ † ( α ) = D ^ † ( α ) D ^ ( α ) = 1 ^ , where 1 ^ is the identity operator. Since D ^ † ( α ) = D ^ ( − α ) , the hermitian conjugate of the displacement operator can also be interpreted as a displacement of opposite magnitude ( − α ). The effect of applying this operator in a similarity transformation of the ladder operators results in their displacement.
D ^ † ( α ) a ^ D ^ ( α ) = a ^ + α D ^ ( α ) a ^ D ^ † ( α ) = a ^ − α The product of two displacement operators is another displacement operator, apart from a phase factor, has the total displacement as the sum of the two individual displacements. This can be seen by utilizing the Baker-Campbell-Hausdorff formula.
e α a ^ † − α ∗ a ^ e β a ^ † − β ∗ a ^ = e ( α + β ) a ^ † − ( β ∗ + α ∗ ) a ^ e ( α β ∗ − α ∗ β ) / 2 . which shows us that:
D ^ ( α ) D ^ ( β ) = e ( α β ∗ − α ∗ β ) / 2 D ^ ( α + β ) When acting on an eigenket, the phase factor e ( α β ∗ − α ∗ β ) / 2 appears in each term of the resulting state, which makes it physically irrelevant.
Two alternative ways to express the displacement operator are:
D ^ ( α ) = e − 1 2 | α | 2 e + α a ^ † e − α ∗ a ^ D ^ ( α ) = e + 1 2 | α | 2 e − α ∗ a ^ e + α a ^ † The displacement operator can also be generalized to multimode displacement. A multimode creation operator can be defined as
A ^ ψ † = ∫ d k ψ ( k ) a ^ † ( k ) ,
where k is the wave vector and its magnitude is related to the frequency ω k according to | k | = ω k / c . Using this definition, we can write the multimode displacement operator as
D ^ ψ ( α ) = exp ( α A ^ ψ † − α ∗ A ^ ψ ) ,
and define the multimode coherent state as
| α ψ ⟩ ≡ D ^ ψ ( α ) | 0 ⟩ .
