Displacement operator

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

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

where $\alpha$ is the amount of displacement in optical phase space, $\alpha ^{\ast }$ is the complex conjugate of that displacement, and ${\hat {a}}$ and ${\hat {a}}^{\dagger }$ 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 $\alpha$. It may also act on the vacuum state by displacing it into a coherent state. Specifically, ${\hat {D}}(\alpha )|0\rangle =|\alpha \rangle$ where $|\alpha \rangle$ is a coherent state. Displaced states are eigenfunctions of the annihilation (lowering) operator.

Properties

The displacement operator is a unitary operator, and therefore obeys ${\hat {D}}(\alpha ){\hat {D}}^{\dagger }(\alpha )={\hat {D}}^{\dagger }(\alpha ){\hat {D}}(\alpha )=I$, where I is the identity matrix. Since ${\hat {D}}^{\dagger }(\alpha )={\hat {D}}(-\alpha )$, 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.

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

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.

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

which shows us that:

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

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

Alternative expressions

Two alternative ways to express the displacement operator are:

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

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

Multimode displacement

The displacement operator can also be generalized to multimode displacement.

See also

• Optical Phase Space