Displacement operator: Difference between revisions

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{{Short description|Mathematical operator in quantum optics}}

{{Context|date=October 2009}}

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 ''α''<math>\alpha</math> is the amount of displacement in [[optical phase space]], ''α''<supmath>\alpha^*</supmath> is the complex conjugate of that displacement, and ''â''<math>\hat{a}</math> and ''â''<supmath>†\hat{a}^\dagger</supmath> are the [[creation and annihilation operators|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 ''α''<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>

The '''displacement operator''' for one mode in [[quantum optics]] is the [[Operator (mathematics)|operator]]

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

where ''α'' is the amount of displacement in [[optical phase space]], ''α''^* is the complex conjugate of that displacement, and ''â'' and ''â''^† are the [[creation and annihilation operators|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 ''α''. 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 |''α''⟩ is a [[coherent state]].

Displaced states are [[eigenfunctions]] 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)=I\hat{1}</math>,

where I<math>\hat{1}</math> is the identity matrixoperator. 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 [[matrix similarity|similarity transformation]] of the ladder operators results in their displacement.

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

:<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, apartup fromto a phase factor, has the total displacement asis the sum of the two individual displacements. This can be seen by utilizing the [[Baker-Campbell-Hausdorff formula#The Hadamard lemma|Baker-Campbell-HausdorffBaker–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^{(\beta\alpha\beta^*-\alpha\beta^*\beta)/2}. </math>

which shows us that:

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:<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^{(\beta\alpha\beta^*-\alpha\beta^*\beta)/2}</math> appears in each term of the resulting state, which makes it physically irrelevant.<ref>Christopher Gerry, Christopher, 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 ==

TwoThe Kermack–McCrea identity (named after [[William Ogilvy Kermack]] and [[William McCrea (astronomer)|William McCrea]]) gives two alternative ways to express the displacement operator are:

:<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\textbfmathbf{k}\psi(\textbfmathbf{k})\hat a^{\dagger}(\textbfmathbf{k})^{\dagger}</math>,

where <math>\textbfmathbf{k}</math> is the wave vector and its magnitude is related to the frequency <math>\omega_{\textbfmathbf{k}}</math> according to <math>|\textbfmathbf{k}|=\omega_{\textbfmathbf{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>,

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

==See also==

* [[Optical Phasephase Spacespace]]

==References==

<references />

{{Physics operators}}

==Notes==

{{Empty section|date=July 2010}}

==See also==

* [[Optical Phase Space]]

[[Category:Quantum optics]]

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