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

D ^ ( α ) = exp ⁡ ( α a ^ † − α ∗ a ^ ) {\displaystyle {\hat {D}}(\alpha )=\exp \left(\alpha {\hat {a}}^{\dagger }-\alpha ^{\ast }{\hat {a}}\right)},

where α {\displaystyle \alpha } is the amount of displacement in optical phase space, α ∗ {\displaystyle \alpha ^{*}} is the complex conjugate of that displacement, and a ^ {\displaystyle {\hat {a}}} and a ^ † {\displaystyle {\hat {a}}^{\dagger }} 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 α {\displaystyle \alpha }. It may also act on the vacuum state by displacing it into a coherent state. Specifically, D ^ ( α ) | 0 ⟩ = | α ⟩ {\displaystyle {\hat {D}}(\alpha )|0\rangle =|\alpha \rangle } where | α ⟩ {\displaystyle |\alpha \rangle } 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.

Properties

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

D ^ † ( α ) a ^ D ^ ( α ) = a ^ + α {\displaystyle {\hat {D}}^{\dagger }(\alpha ){\hat {a}}{\hat {D}}(\alpha )={\hat {a}}+\alpha }

D ^ ( α ) a ^ D ^ † ( α ) = a ^ − α {\displaystyle {\hat {D}}(\alpha ){\hat {a}}{\hat {D}}^{\dagger }(\alpha )={\hat {a}}-\alpha }

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.

e α a ^ † − α ∗ a ^ e β a ^ † − β ∗ a ^ = e ( α + β ) a ^ † − ( β ∗ + α ∗ ) a ^ e ( α β ∗ − α ∗ β ) / 2 . {\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}.}

which shows us that:

D ^ ( α ) D ^ ( β ) = e ( α β ∗ − α ∗ β ) / 2 D ^ ( α + β ) {\displaystyle {\hat {D}}(\alpha ){\hat {D}}(\beta )=e^{(\alpha \beta ^{*}-\alpha ^{*}\beta )/2}{\hat {D}}(\alpha +\beta )}

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

It further leads to the braiding relation

D ^ ( α ) D ^ ( β ) = e α β ∗ − α ∗ β D ^ ( β ) D ^ ( α ) {\displaystyle {\hat {D}}(\alpha ){\hat {D}}(\beta )=e^{\alpha \beta ^{*}-\alpha ^{*}\beta }{\hat {D}}(\beta ){\hat {D}}(\alpha )}

Alternative expressions

The Kermack–McCrea identity (named after William Ogilvy Kermack and William McCrea) gives two alternative ways to express the displacement operator:

D ^ ( α ) = e − 1 2 | α | 2 e + α a ^ † e − α ∗ a ^ {\displaystyle {\hat {D}}(\alpha )=e^{-{\frac {1}{2}}|\alpha |^{2}}e^{+\alpha {\hat {a}}^{\dagger }}e^{-\alpha ^{*}{\hat {a}}}}

D ^ ( α ) = e + 1 2 | α | 2 e − α ∗ a ^ e + α a ^ † {\displaystyle {\hat {D}}(\alpha )=e^{+{\frac {1}{2}}|\alpha |^{2}}e^{-\alpha ^{*}{\hat {a}}}e^{+\alpha {\hat {a}}^{\dagger }}}

In the Cahill-Glauber s {\displaystyle s}-order represntation we can write some useful definitions of these forms of the displacement operator.

D ^ symmetric ( α ) ≡ D ^ 0 ( α ) ≡ D ^ ( α ) = e + α a ^ † − α ∗ a ^ {\displaystyle {\hat {D}}_{\text{symmetric}}(\alpha )\equiv {\hat {D}}_{0}(\alpha )\equiv {\hat {D}}(\alpha )=e^{+\alpha {\hat {a}}^{\dagger }-\alpha ^{*}{\hat {a}}}}

D ^ normal ( α ) ≡ D ^ + 1 ( α ) ≡ e + α a ^ † e − α ∗ a ^ {\displaystyle {\hat {D}}_{\text{normal}}(\alpha )\equiv {\hat {D}}_{+1}(\alpha )\equiv e^{+\alpha {\hat {a}}^{\dagger }}e^{-\alpha ^{*}{\hat {a}}}}

D ^ anti-normal ( α ) ≡ D ^ − 1 ( α ) ≡ e − α ∗ a ^ e + α a ^ † {\displaystyle {\hat {D}}_{\text{anti-normal}}(\alpha )\equiv {\hat {D}}_{-1}(\alpha )\equiv e^{-\alpha ^{*}{\hat {a}}}e^{+\alpha {\hat {a}}^{\dagger }}}

With the generalization:

D ^ s ( α ) ≡ D 0 ^ ( α ) e s 2 | α | 2 = e + α a ^ † − α ∗ a ^ e s 2 | α | 2 {\displaystyle {\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}}}

Relationship to the Symmetric Delta Function

The displacement operator is the fourier transform of the symmetric delta function

T ^ 0 ( α ) ≡ π δ 0 ( 2 ) ( a ^ − α , a ^ † − α ∗ ) = ∫ d 2 β π D ^ 0 ( β ) e β ∗ α − β α ∗ {\displaystyle {\hat {T}}_{0}(\alpha )\equiv \pi \delta _{0}^{(2)}({\hat {a}}-\alpha ,{\hat {a}}^{\dagger }-\alpha ^{*})=\int {\frac {d^{2}\beta }{\pi }}{\hat {D}}_{0}(\beta )e^{\beta ^{*}\alpha -\beta \alpha ^{*}}}

This is extended to the generally ordered delta function:

T ^ s ( α ) ≡ π δ s ( 2 ) ( a ^ − α , a ^ † − α ∗ ) = ∫ d 2 β π D ^ s ( β ) e β ∗ α − β α ∗ {\displaystyle {\hat {T}}_{s}(\alpha )\equiv \pi \delta _{s}^{(2)}({\hat {a}}-\alpha ,{\hat {a}}^{\dagger }-\alpha ^{*})=\int {\frac {d^{2}\beta }{\pi }}{\hat {D}}_{s}(\beta )e^{\beta ^{*}\alpha -\beta \alpha ^{*}}}

Example: Normal ordered delta function

T ^ + 1 ( α ) = ∫ d 2 β π D ^ + 1 ( β ) e β ∗ α − β α ∗ = ∫ d 2 β π e a ^ † β e − a ^ β ∗ e β ∗ α − β α ∗ = ∫ d 2 β π e ( a ^ † − α ∗ ) β e ( α − a ^ ) β ∗ = 1 π [ π δ ( 1 ) ( a ^ † − α ∗ ) ] [ π δ ( 1 ) ( a ^ − α ) ] = π δ + 1 ( 2 ) ( a ^ − α , a ^ † − α ∗ ) {\displaystyle {\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 _{+1}^{(2)}({\hat {a}}-\alpha ,{\hat {a}}^{\dagger }-\alpha ^{*})\end{aligned}}}

Multimode displacement

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

A ^ ψ † = ∫ d k ψ ( k ) a ^ † ( k ) {\displaystyle {\hat {A}}_{\psi }^{\dagger }=\int d\mathbf {k} \psi (\mathbf {k} ){\hat {a}}^{\dagger }(\mathbf {k} )},

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

D ^ ψ ( α ) = exp ⁡ ( α A ^ ψ † − α ∗ A ^ ψ ) {\displaystyle {\hat {D}}_{\psi }(\alpha )=\exp \left(\alpha {\hat {A}}_{\psi }^{\dagger }-\alpha ^{\ast }{\hat {A}}_{\psi }\right)},

and define the multimode coherent state as

| α ψ ⟩ ≡ D ^ ψ ( α ) | 0 ⟩ {\displaystyle |\alpha _{\psi }\rangle \equiv {\hat {D}}_{\psi }(\alpha )|0\rangle }.

See also