Displacement operator explained

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 \left ( \alpha \hat{a} ^{\dagger} - \alpha ^{\ast} \hat{a} \right ) ,where \alpha is the amount of displacement in optical phase space, \alpha ^{*} 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, which is an eigenstate 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)=\hat{1}} ,where \hat{ 1 } is the identity operator. 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 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} ^{\dagger} - \alpha ^{*\hat{a}}} e ^{\beta\hat{a} ^{\dagger} - \beta ^{*\hat{a}}} = e ^{(\alpha} ^{\dagger} - (\beta ^{*+\alpha} ^{*)\hat{a}}} e ^{(\alpha\beta ^{*-\alpha} ^{*\beta)/2}} .

which shows us that:

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

When acting on an eigenket, the phase factor

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

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

Alternative expressions

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

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

Multimode displacement

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

\hat A ^{\dagger} _{\psi} = \int d k \psi( k )\hat a ^{\dagger} ( 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 \left ( \alpha \hat A ^{\dagger} _{\psi} - \alpha ^{\ast} \hat A _{\psi} \right ) ,

and define the multimode coherent state as

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

See also

• Optical phase space

References

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