Displacement operator

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

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

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, ${\displaystyle {\hat {D}}(\alpha )|0\rangle =|\alpha \rangle }$ where |α⟩ is a coherent state. Displaced states are eigenfunctions of the annihilation (lowering) operator.

Properties

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

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

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 ${\displaystyle e^{(\beta \alpha ^{*}-\alpha \beta ^{*})/2}}$  appears in each term of the resulting state, which makes it physically irrelevant.^[1]

Alternative expressions

Two alternative ways to express the displacement operator are:

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

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

Multimode displacement

The displacement operator can also be generalized to multimode displacement.

References

1. Gerry, Christopher, and Peter Knight: Introductory Quantum Optics. Cambridge (England): Cambridge UP, 2005.

Notes

See also

• Optical Phase Space