# Displacement operator

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{{Short description|Mathematical operator in quantum optics}}
In the [quantum mechanics](/source/quantum_mechanics) study of [optical phase space](/source/optical_phase_space),  the '''displacement operator''' for one mode is the [shift operator](/source/shift_operator)  in [quantum optics](/source/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](/source/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](/source/creation_and_annihilation_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](/source/coherent_state). Specifically,
<math>\hat{D}(\alpha)|0\rangle=|\alpha\rangle</math> where <math>|\alpha\rangle</math> is a [coherent state](/source/coherent_state), which is an [eigenstate](/source/eigenstate) of the annihilation (lowering) operator. This operator was introduced independently by [Richard Feynman](/source/Richard_Feynman) and [Roy J. Glauber](/source/Roy_J._Glauber) in 1951.<ref>{{Cite journal |last=Dodonov |first=V. V. |date=2002 |title='Nonclassical' states in quantum optics: a 'squeezed' review of the first 75 years |url=https://iopscience.iop.org/article/10.1088/1464-4266/4/1/201 |journal=Journal of Optics B: Quantum and Semiclassical Optics |volume=4 |issue=1}}</ref><ref>{{Cite journal |last=Feynman |first=Richard P. |date=1951-10-01 |title=An Operator Calculus Having Applications in Quantum Electrodynamics |url=https://journals.aps.org/pr/abstract/10.1103/PhysRev.84.108 |journal=Physical Review |volume=84 |issue=1 |pages=108–128 |doi=10.1103/PhysRev.84.108|url-access=subscription }}</ref><ref>{{Cite journal |last=Glauber |first=Roy J. |date=1951-11-01 |title=Some Notes on Multiple-Boson Processes |url=https://journals.aps.org/pr/abstract/10.1103/PhysRev.84.395 |journal=Physical Review |volume=84 |issue=3 |pages=395–400 |doi=10.1103/PhysRev.84.395|url-access=subscription }}</ref>

== Properties ==
The displacement operator is a [unitary operator](/source/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](/source/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](/source/matrix_similarity) 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](/source/Baker%E2%80%93Campbell%E2%80%93Hausdorff_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.<ref>Christopher Gerry and Peter Knight: ''Introductory Quantum Optics''. Cambridge (England): Cambridge UP, 2005.</ref>

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–McCrea identity (named after [William Ogilvy Kermack](/source/William_Ogilvy_Kermack) and [William McCrea](/source/William_McCrea_(astronomer))) 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>

In the Cahill-Glauber <math>s</math>-order representation we can write some useful definitions of these forms of the displacement operator.
:<math> \hat{D}_{\text{symmetric}}(\alpha) 
\equiv \hat{D}_{0}(\alpha) 
\equiv \hat{D}(\alpha) 
= e^{+\alpha \hat{a}^{\dagger} -\alpha^{*} \hat{a} }</math>
:<math> \hat{D}_{\text{normal}}(\alpha) 
\equiv \hat{D}_{+1}(\alpha) 
\equiv e^{+\alpha \hat{a}^{\dagger}} e^{-\alpha^{*} \hat{a} }</math>
:<math>\hat{D}_{\text{anti-normal}}(\alpha) 
\equiv \hat{D}_{-1}(\alpha) 
\equiv e^{-\alpha^{*} \hat{a} }e^{+\alpha \hat{a}^{\dagger}} </math>

With the generalization: <ref>{{cite journal
 |last1=Cahill
 |first1=K. E.
 |last2=Glauber
 |first2=R. J.
 |title=Density Operators and Quasiprobability Distributions
 |journal=Physical Review
 |volume=177
 |issue=5
 |pages=1882–1902
 |date=1969-01-25
 |doi=10.1103/physrev.177.1882
}}</ref>
:<math> \hat{D}_{s}(\alpha) 
\equiv \hat{D_0}(\alpha) e^{\frac{s}{2}|\alpha|^2}
= e^{+\alpha \hat{a}^{\dagger} -\alpha^{*} \hat{a} } e^{\frac{s}{2}|\alpha|^2}</math>

== Relationship to the Symmetric Delta Function ==
The displacement operator is the [fourier transform](/source/fourier_transform) of the symmetric delta function 
:<math>\hat{T}_0 (\alpha) 
\equiv \pi \delta^{(2)}_0(\hat{a}-\alpha, \hat{a}^\dagger - \alpha^*)
= \int \frac{d^2 \beta}{\pi} \hat{D}_0(\beta) e^{\beta^* \alpha - \beta \alpha^*}
</math>

This is extended to the generally ordered delta function: <ref>{{cite journal
 |last1=Cahill
 |first1=K. E.
 |last2=Glauber
 |first2=R. J.
 |title=Density Operators and Quasiprobability Distributions
 |journal=Physical Review
 |volume=177
 |issue=5
 |pages=1882–1902
 |date=1969-01-25
 |doi=10.1103/physrev.177.1882
}}</ref>
:<math>\hat{T}_s (\alpha) \equiv \pi \delta^{(2)}_s(\hat{a}-\alpha, \hat{a}^\dagger - \alpha^*)
= \int \frac{d^2 \beta}{\pi} \hat{D}_s(\beta) e^{\beta^* \alpha - \beta \alpha^*}
</math>

<strong>Example: [Normal order](/source/Normal_order)ed delta function </strong>
:<math>
\begin{aligned}
\hat{T}_{+1}(\alpha)
&= \int \frac{d^2 \beta}{\pi} \hat{D}_{+1}(\beta) e^{\beta^* \alpha - \beta \alpha^*}\\
&= \int \frac{d^2 \beta}{\pi} e^{\hat{a}^\dagger \beta} e^{-\hat{a} \beta^*} e^{\beta^* \alpha - \beta \alpha^*} \\
&= \int \frac{d^2 \beta}{\pi} e^{(\hat{a}^\dagger - \alpha^*) \beta} e^{(\alpha-\hat{a}) \beta^*}  \\
&= \frac{1}{\pi} \left[ \pi \delta^{(1)}(\hat{a}^\dagger - \alpha^*) \right] \left[ \pi \delta^{(1)}(\hat{a} - \alpha) \right] \\
&= \pi \delta^{(2)}_{+1}(\hat{a}-\alpha, \hat{a}^\dagger - \alpha^*)
\end{aligned}
</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](/source/Optical_phase_space)
==References==
<references />

{{Physics operators}}

Category:Quantum optics

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Adapted from the Wikipedia article [Displacement operator](https://en.wikipedia.org/wiki/Displacement_operator) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Displacement_operator?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.