Constraint algebra

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

$\hat{D}(\alpha )=\exp (\alpha {\hat{a}}^{†}-{\alpha }^{\ast }\hat{a})$,

where α is the amount of displacement in optical phase space, α^* is the complex conjugate of that displacement, and â and â^† 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, $\hat{D}(\alpha )|0⟩=|\alpha ⟩$ where $|\alpha ⟩$ is a coherent state. Displaced states are eigenfunctions of the annihilation (lowering) operator.

Properties

The displacement operator is a unitary operator, and therefore obeys $\hat{D}(\alpha ){\hat{D}}^{†}(\alpha )={\hat{D}}^{†}(\alpha )\hat{D}(\alpha )=I$, where I is the identity matrix. Since ${\hat{D}}^{†}(\alpha )=\hat{D}(-\alpha )$, the hermitian conjugate of the displacement operator can also be interpreted as a displacement of opposite magnitude ($-\alpha$). The effect of applying this operator in a similarity transformation of the ladder operators results in their displacement.

${\hat{D}}^{†}(\alpha )\hat{a}\hat{D}(\alpha )=\hat{a}+\alpha$

$\hat{D}(\alpha )\hat{a}{\hat{D}}^{†}(\alpha )=\hat{a}-\alpha$

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.

$e^{\alpha {\hat{a}}^{†}-{\alpha }^{\ast }\hat{a}}{e}^{\beta {\hat{a}}^{†}-{\beta }^{\ast }\hat{a}}=e^{(\alpha +\beta ){\hat{a}}^{†}-({\beta }^{\ast }+{\alpha }^{\ast })\hat{a}}{e}^{(\alpha {\beta }^{\ast }-{\alpha }^{\ast }\beta )/2}.$

which shows us that:

$\hat{D}(\alpha )\hat{D}(\beta )=e^{(\alpha {\beta }^{\ast }-{\alpha }^{\ast }\beta )/2}\hat{D}(\alpha +\beta )$

When acting on an eigenket, the phase factor $e^{(\alpha {\beta }^{\ast }-{\alpha }^{\ast }\beta )/2}$ appears in each term of the resulting state, which makes it physically irrelevant.

Alternative expressions

Two alternative ways to express the displacement operator are:

$\hat{D}(\alpha )=e^{-\frac{1}{2}|\alpha {|}^2}{e}^{+\alpha {\hat{a}}^{†}}{e}^{-{\alpha }^{\ast }\hat{a}}$

$\hat{D}(\alpha )=e^{+\frac{1}{2}|\alpha {|}^2}{e}^{-{\alpha }^{\ast }\hat{a}}{e}^{+\alpha {\hat{a}}^{†}}$

Multimode displacement

The displacement operator can also be generalized to multimode displacement. A multimode creation operator can be defined as

${\hat{A}}_{\psi }^{†}=\int d𝐤\psi (𝐤)\hat{a}(𝐤{)}^{†}$,

where $𝐤$ is the wave vector and its magnitude is related to the frequency ${\omega }_{𝐤}$ according to $|𝐤|={\omega }_{𝐤}/c$. Using this definition, we can write the multimode displacement operator as

${\hat{D}}_{\psi }(\alpha )=\exp (\alpha {\hat{A}}_{\psi }^{†}-{\alpha }^{\ast }{\hat{A}}_{\psi })$,

and define the multimode coherent state as

$|{\alpha }_{\psi }⟩\equiv {\hat{D}}_{\psi }(\alpha )|0⟩$.

See also

• Optical Phase Space