Physics:Displacement operator

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

[math]\displaystyle{ \hat{D}(\alpha)=\exp \left ( \alpha \hat{a}^\dagger - \alpha^\ast \hat{a} \right ) }[/math],

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

Properties

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

[math]\displaystyle{ \hat{D}^\dagger(\alpha) \hat{a} \hat{D}(\alpha)=\hat{a}+\alpha }[/math]

[math]\displaystyle{ \hat{D}(\alpha) \hat{a} \hat{D}^\dagger(\alpha)=\hat{a}-\alpha }[/math]

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.

[math]\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}. }[/math]

which shows us that:

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

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

It further leads to the braiding relation

[math]\displaystyle{ \hat{D}(\alpha)\hat{D}(\beta)=e^{\alpha\beta^*-\alpha^*\beta} \hat{D}(\beta)\hat{D}(\alpha) }[/math]

Alternative expressions

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

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

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

Multimode displacement

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

[math]\displaystyle{ \hat A_{\psi}^{\dagger}=\int d\mathbf{k}\psi(\mathbf{k})\hat a^{\dagger}(\mathbf{k}) }[/math],

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

[math]\displaystyle{ \hat{D}_{\psi}(\alpha)=\exp \left ( \alpha \hat A_{\psi}^{\dagger} - \alpha^\ast \hat A_{\psi} \right ) }[/math],

and define the multimode coherent state as

[math]\displaystyle{ |\alpha_{\psi}\rangle\equiv\hat{D}_{\psi}(\alpha)|0\rangle }[/math].

See also

• Optical phase space