{{Short description|Mathematical operator in quantum optics}} 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. This operator was introduced independently by Richard Feynman and 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, 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.<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 and William McCrea) 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 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 ordered 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 ==References== <references />

{{Physics operators}}

Category:Quantum optics