Английская Википедия:Displacement operator

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

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

where <math>\alpha</math> is the amount of displacement in optical phase space, <math>\alpha^*</math> is the complex conjugate of that displacement, and <math>\hat{a}</math> and <math>\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>\alpha</math>. It may also act on the vacuum state by displacing it into a coherent state. Specifically, <math>\hat{D}(\alpha)|0\rangle=|\alpha\rangle</math> where <math>|\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>\hat{D}(\alpha)\hat{D}^\dagger(\alpha)=\hat{D}^\dagger(\alpha)\hat{D}(\alpha)=\hat{1}</math>, where <math>\hat{1}</math> is the identity operator. Since <math> \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>-\alpha</math>). The effect of applying this operator in a similarity transformation of the ladder operators results in their displacement.

<math>\hat{D}^\dagger(\alpha) \hat{a} \hat{D}(\alpha)=\hat{a}+\alpha</math>

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

<math>\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>\hat A_{\psi}^{\dagger}=\int d\mathbf{k}\psi(\mathbf{k})\hat a^{\dagger}(\mathbf{k})</math>,

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

<math>\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>|\alpha_{\psi}\rangle\equiv\hat{D}_{\psi}(\alpha)|0\rangle</math>.

See also

• Optical phase space