Displacement operator

Template:Short description In the quantum mechanics study of optical phase space, the displacement operator for one mode is the shift operator in quantum optics,

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

where $\alpha$ is the amount of displacement in optical phase space, ${\alpha }^{\ast }$ is the complex conjugate of that displacement, and $\hat{a}$ and ${\hat{a}}^{†}$ 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 $\alpha$. 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, which is an eigenstate of the annihilation (lowering) operator. This operator was introduced independently by Richard Feynman and Roy J. Glauber in 1951.

Properties

The displacement operator is a unitary operator, and therefore obeys $\hat{D}(\alpha ){\hat{D}}^{†}(\alpha )={\hat{D}}^{†}(\alpha )\hat{D}(\alpha )=\hat{1}$, where $\hat{1}$ is the identity operator. 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 whose total displacement, up to a phase factor, is 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.

It further leads to the braiding relation

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

Alternative expressions

The Kermack–McCrea identity (named after William Ogilvy Kermack and William McCrea) gives two alternative ways to express the displacement operator:

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

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

In the Cahill-Glauber $s$-order representation we can write some useful definitions of these forms of the displacement operator.

${\hat{D}}_{\text{symmetric}}(\alpha )\equiv {\hat{D}}_0(\alpha )\equiv \hat{D}(\alpha )=e^{+\alpha {\hat{a}}^{†}-{\alpha }^{\ast }\hat{a}}$

${\hat{D}}_{\text{normal}}(\alpha )\equiv {\hat{D}}_{+1}(\alpha )\equiv e^{+\alpha {\hat{a}}^{†}}{e}^{-{\alpha }^{\ast }\hat{a}}$

${\hat{D}}_{\text{anti-normal}}(\alpha )\equiv {\hat{D}}_{-1}(\alpha )\equiv e^{-{\alpha }^{\ast }\hat{a}}{e}^{+\alpha {\hat{a}}^{†}}$

With the generalization:

${\hat{D}}_s(\alpha )\equiv \widehat{D_0}(\alpha )e^{\frac{s}{2}|\alpha {|}^2}=e^{+\alpha {\hat{a}}^{†}-{\alpha }^{\ast }\hat{a}}{e}^{\frac{s}{2}|\alpha {|}^2}$

Relationship to the Symmetric Delta Function

The displacement operator is the fourier transform of the symmetric delta function

${\hat{T}}_0(\alpha )\equiv \pi {\delta }_0^{(2)}(\hat{a}-\alpha ,{\hat{a}}^{†}-{\alpha }^{\ast })=\int \frac{d^2\beta }{\pi }{\hat{D}}_0(\beta )e^{{\beta }^{\ast }\alpha -\beta {\alpha }^{\ast }}$

This is extended to the generally ordered delta function:

${\hat{T}}_s(\alpha )\equiv \pi {\delta }_s^{(2)}(\hat{a}-\alpha ,{\hat{a}}^{†}-{\alpha }^{\ast })=\int \frac{d^2\beta }{\pi }{\hat{D}}_s(\beta )e^{{\beta }^{\ast }\alpha -\beta {\alpha }^{\ast }}$

Example: Normal ordered delta function

$\begin{array}{cc}{\hat{T}}_{+1}(\alpha )& =\int \frac{d^2\beta }{\pi }{\hat{D}}_{+1}(\beta )e^{{\beta }^{\ast }\alpha -\beta {\alpha }^{\ast }}\\ & =\int \frac{d^2\beta }{\pi }{e}^{{\hat{a}}^{†}\beta }{e}^{-\hat{a}{\beta }^{\ast }}{e}^{{\beta }^{\ast }\alpha -\beta {\alpha }^{\ast }}\\ & =\int \frac{d^2\beta }{\pi }{e}^{({\hat{a}}^{†}-{\alpha }^{\ast })\beta }{e}^{(\alpha -\hat{a}){\beta }^{\ast }}\\ & =\frac{1}{\pi }[\pi {\delta }^{(1)}({\hat{a}}^{†}-{\alpha }^{\ast })][\pi {\delta }^{(1)}(\hat{a}-\alpha )]\\ & =\pi {\delta }_{+1}^{(2)}(\hat{a}-\alpha ,{\hat{a}}^{†}-{\alpha }^{\ast })\end{array}$

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