Displacement operator

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

${\displaystyle {\hat {D}}(\alpha )=\exp \left(\alpha {\hat {a}}^{\dagger }-\alpha ^{\ast }{\hat {a}}\right)}$,

where ${\displaystyle \alpha }$ is the amount of displacement in optical phase space, ${\displaystyle \alpha ^{*}}$ is the complex conjugate of that displacement, and ${\displaystyle {\hat {a}}}$ and ${\displaystyle {\hat {a}}^{\dagger }}$ 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 ${\displaystyle \alpha }$. It may also act on the vacuum state by displacing it into a coherent state. Specifically, ${\displaystyle {\hat {D}}(\alpha )|0\rangle =|\alpha \rangle }$ where ${\displaystyle |\alpha \rangle }$ 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.^[1][2][3]

The displacement operator is a unitary operator, and therefore obeys ${\displaystyle {\hat {D}}(\alpha ){\hat {D}}^{\dagger }(\alpha )={\hat {D}}^{\dagger }(\alpha ){\hat {D}}(\alpha )={\hat {1}}}$, where ${\displaystyle {\hat {1}}}$ is the identity operator. Since ${\displaystyle {\hat {D}}^{\dagger }(\alpha )={\hat {D}}(-\alpha )}$, the hermitian conjugate of the displacement operator can also be interpreted as a displacement of opposite magnitude (${\displaystyle -\alpha }$). The effect of applying this operator in a similarity transformation of the ladder operators results in their displacement.

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

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

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.

${\displaystyle e^{\alpha {\hat {a}}^{\dagger }-\alpha ^{*}{\hat {a}}}e^{\beta {\hat {a}}^{\dagger }-\beta ^{*}{\hat {a}}}=e^{(\alpha +\beta ){\hat {a}}^{\dagger }-(\beta ^{*}+\alpha ^{*}){\hat {a}}}e^{(\alpha \beta ^{*}-\alpha ^{*}\beta )/2}.}$

which shows us that:

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

When acting on an eigenket, the phase factor ${\displaystyle e^{(\alpha \beta ^{*}-\alpha ^{*}\beta )/2}}$ appears in each term of the resulting state, which makes it physically irrelevant.^[4]

It further leads to the braiding relation

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

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

${\displaystyle {\hat {D}}(\alpha )=e^{-{\frac {1}{2}}|\alpha |^{2}}e^{+\alpha {\hat {a}}^{\dagger }}e^{-\alpha ^{*}{\hat {a}}}}$

${\displaystyle {\hat {D}}(\alpha )=e^{+{\frac {1}{2}}|\alpha |^{2}}e^{-\alpha ^{*}{\hat {a}}}e^{+\alpha {\hat {a}}^{\dagger }}}$

In the Cahill-Glauber ${\displaystyle s}$-order represntation we can write some useful definitions of these forms of the displacement operator.

${\displaystyle {\hat {D}}_{\text{symmetric}}(\alpha )\equiv {\hat {D}}_{0}(\alpha )\equiv {\hat {D}}(\alpha )=e^{+\alpha {\hat {a}}^{\dagger }-\alpha ^{*}{\hat {a}}}}$

${\displaystyle {\hat {D}}_{\text{normal}}(\alpha )\equiv {\hat {D}}_{+1}(\alpha )\equiv e^{+\alpha {\hat {a}}^{\dagger }}e^{-\alpha ^{*}{\hat {a}}}}$

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

With the generalization: ^[5]

${\displaystyle {\hat {D}}_{s}(\alpha )\equiv {\hat {D_{0}}}(\alpha )e^{{\frac {s}{2}}|\alpha |^{2}}=e^{+\alpha {\hat {a}}^{\dagger }-\alpha ^{*}{\hat {a}}}e^{{\frac {s}{2}}|\alpha |^{2}}}$

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

${\displaystyle {\hat {T}}_{0}(\alpha )\equiv \pi \delta _{0}^{(2)}({\hat {a}}-\alpha ,{\hat {a}}^{\dagger }-\alpha ^{*})=\int {\frac {d^{2}\beta }{\pi }}{\hat {D}}_{0}(\beta )e^{\beta ^{*}\alpha -\beta \alpha ^{*}}}$

This is extended to the generally ordered delta function: ^[6]

${\displaystyle {\hat {T}}_{s}(\alpha )\equiv \pi \delta _{s}^{(2)}({\hat {a}}-\alpha ,{\hat {a}}^{\dagger }-\alpha ^{*})=\int {\frac {d^{2}\beta }{\pi }}{\hat {D}}_{s}(\beta )e^{\beta ^{*}\alpha -\beta \alpha ^{*}}}$

Example: Normal ordered delta function

${\displaystyle {\begin{aligned}{\hat {T}}_{+1}(\alpha )&=\int {\frac {d^{2}\beta }{\pi }}{\hat {D}}_{+1}(\beta )e^{\beta ^{*}\alpha -\beta \alpha ^{*}}\\&=\int {\frac {d^{2}\beta }{\pi }}e^{{\hat {a}}^{\dagger }\beta }e^{-{\hat {a}}\beta ^{*}}e^{\beta ^{*}\alpha -\beta \alpha ^{*}}\\&=\int {\frac {d^{2}\beta }{\pi }}e^{({\hat {a}}^{\dagger }-\alpha ^{*})\beta }e^{(\alpha -{\hat {a}})\beta ^{*}}\\&={\frac {1}{\pi }}\left[\pi \delta ^{(1)}({\hat {a}}^{\dagger }-\alpha ^{*})\right]\left[\pi \delta ^{(1)}({\hat {a}}-\alpha )\right]\\&=\pi \delta _{+1}^{(2)}({\hat {a}}-\alpha ,{\hat {a}}^{\dagger }-\alpha ^{*})\end{aligned}}}$

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

${\displaystyle {\hat {A}}_{\psi }^{\dagger }=\int d\mathbf {k} \psi (\mathbf {k} ){\hat {a}}^{\dagger }(\mathbf {k} )}$,

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

${\displaystyle {\hat {D}}_{\psi }(\alpha )=\exp \left(\alpha {\hat {A}}_{\psi }^{\dagger }-\alpha ^{\ast }{\hat {A}}_{\psi }\right)}$,

and define the multimode coherent state as

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

• Optical phase space