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.

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.^[1]

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-McCrae identity gives two alternative ways to express the displacement operator:

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

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

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