As a generalization of the Fourier transform, the fractional Fourier transform (FrFT) has attracted much attention and is used widely to design optical systems and optimize holographic storage efficiency^[1–5], whose applications range from filter design and signal analysis to phase retrieval and pattern recognition. Is there correspondence between the classical-optical transform and quantum-optical transform (unitary operator)? By using the Dirac’s symbolic method and the technique of integration within an ordered product (IWOP), the corresponding relation between them has been bridged^[6]. For example, the wavelet transform is transformed into a matrix element of a squeezing-translating operator between the state vector corresponding to the given mother wavelet and the state to be transformed by using this method^[7]. This indicates that we may not only find a new quantum mechanical unitary operator corresponding to the classical transform, but we also propose some new classical-optical transforms from the point of quantum mechanics. In this Letter, on the basis of complex FrFT, we shall introduce a new FrET in a 2D case, whose quantum mechanical operator corresponds to an entangling operator.

A brief review of the FrFT and complex FrFT is presented next. The FrFT in one dimension of θ order is defined as Fθ[f(x)]=∫−∞∞Kθ(x,y)f(x)dx,(1)where the integral kernel function Kθ(x,y) is Kθ(x,y)=ei(π2−θ)2πsinθexp{−ix2+y22tanθ+ixysinθ}.(2)

If we define f(x)≡〈x|f〉 and Kθ(x,y)=〈y|Sθ|x〉 where |y〉 and |x〉 are coordinate Eigenvectors, X|x〉=x|x〉, then multiplying the function Kθ(x,y) by the ket ∫−∞∞dy|y〉 and bra ∫−∞∞dx〈x| from left and right, and using the IWOP technique^[8] to perform the integration, we can finally get $$\begin{gathered} Sθ=∫−∞∞dxdy|y〉Kθ(x,y)〈x| \\ ≕exp{(eiθ−1)a†a}: \\ =exp{iθa†a}, \end{gathered}$$(3)where in the last step, we have used the formula exp{fa†a}≕exp{(ef−1)a†a}:^[9,10]. Thus the FrFT can be described, in quantum-mechanical language, as Fθ[f(x)]=∫−∞∞〈y|Sθ|x〉〈x|f〉dx=〈y|Sθ|f〉,(4)where the completeness relation of coordinate representation is used ∫−∞∞dx|x〉〈x|=1.(5)

From Eqs. (3) and (4), one can clearly see that the unitary operator Sθ corresponding to the 1D FrFT is just the rotation operator, and the FrFT is actually the matrix element of Sθ in coordinate state 〈y| and transformed vector |f〉 under the quantum mechanical version of the FrFT. Employing this version, it is easy and direct to discuss some properties of the FrFT.

As an extension, the complex FrFT is introduced by defining the following transform^[11]$$\begin{gathered} Fθ[f](η′)=∫−∞∞d2ηπKC(η′,η)f(η), \\ KC(η′,η)=ei(θ−π2)2sinθexp[i(|η′|2+|η|2)2tanθ−i(η′*η+η*η′)2sinθ]. \end{gathered}$$(6)

In a similar way, using the IWOP technique and the completeness relation of the entangled state representation^[9]∫−∞∞d2ηπ|η〉〈η|=1.(7)where |η=η1+iη2〉 is the Eigenvector of commutable operators: relative position X1−X2 and the total momentum P1+P2, i.e., (X1−X2)|η〉=2η1|η〉, (P1+P2)|η〉=2η2|η〉, and |η〉=exp[−12|η|2+ηa†−η*b†+a†b†]|00〉.(8)We can present Eq. (6) as the quantum version of Fθ[f](η′)=〈η′|exp[−iθ(a†a+b†b)]|f〉.(9)

Actually, from Eqs. (3) and (9) we see that the corresponding unitary operators of the 1D and complex FrFT are the (product of) rotation operators. Then an interesting question naturally arises. Is there a unitary entangling operator corresponding to a kind of complex FrFT? The answer is affirmative. Next we shall propose such a fractional transform.

Regarding the fractional entangling transform, if we replace the integration kernel [Eq. (6)] with KC(η′,η)=12sinθ⁢exp[i(η′2+η2+η′*2+η*2)4tanθ−i(ηη′+η*η′*)2sinθ],(10)then to what kind of transform does it belong? Is it still a fractional FrFT? Actually, Eq. (10) just corresponds to the integration kernel of the fractional entangling transform. Letting KC=〈η′|Uθ|η〉, and using the completeness relation of entangled state representation [Eq. (7)], we can express the operator Uθ as Uθ=∫−∞∞d2ηd2η′π2|η′〉KC(η′,η)〈η|.(11)

Furthermore, use of Eq. (8) and the normally ordering form of vacuum project operator |00〉〈00|≕exp{−a†a−b†b}:^[9,12] as well as the IWOP technique, we can derive $$\begin{gathered} Uθ=12sinθ∫d2ηd2η′π2:exp{−12(|η′|2+|η|2) \\ +η′a†−η′*b†+η*a−ηb+i(η2+η*2+η′2+η′*2)4tanθ \\ −i(ηη′+η*η′*)2sinθ+ab+a†b†−a†a−b†b} \\ ≕exp[(cosθ−1)(a†a+b†b)+i(ab†+a†b)sinθ]: \end{gathered}$$(12)where we have used the integration formula (Appendix A of Ref. [13]) $$\begin{gathered} ∫d2zπexp(−ζ|z|2+ξz+ηz*+fz2+gz*2) \\ =1ζ2−4fgexp[ζξη+ξ2g+η2fζ2−4fg]. \end{gathered}$$(13)

Equation (12) is the normally ordering form of operator Uθ. In order to see clearly the quantum mechanical correspondence of Eq. (12), we first derive the transform relation UθaUθ†. Using the normally ordering form of Uθ and the completeness relation of the coherent state ∫d2α|α〉〈α|/π=1, we can directly derive $$\begin{gathered} UθaUθ†=∂∂τexp[(acosθ−ibsinθ)τ]|τ=0 \\ =acosθ−ibsinθ, \\ UθbUθ†=bcosθ−iasinθ, \end{gathered}$$(14)where we have used the IWOP technique and Eq. (13). Thus, making the partial derivative for parameter θ and using the transform in Eq. (14) we can obtain $$\begin{gathered} ∂∂θUθ=i(b†Uθa+a†Uθb)cosθ−(a†Uθa+b†Uθb)sinθ \\ =[i(b†UθaUθ†+a†UθbUθ†)cosθ−(a†UθaUθ†+b†UθbUθ†)sinθ]Uθ \\ =i(ab†+a†b)Uθ, \end{gathered}$$(15)which indicates that Uθ=exp{iθ(ab†+a†b)}.(16)

Obviously, Uθ is a unitary operator (Uθ†=Uθ−1). It is interesting to note that Uθ is actually a two-mode squeezing operator (or entangling operator). Thus we name the unitary operator Uθ the fractional entangling operator which is evidently different from the one of complex FrFT in Eq. (9). In fact, the unitary operator Uθ can be realized experimentally by the interactions correspond to positive detuning when the resonator frequency is larger than the laser frequency in the optomechanical case (a nonlinear process)^[14].

Regarding the additivity of fractional transform, an important feature of the FrFT is the additivity property, i.e., FθFθ′=Fθ+θ′. In order to prove this property of the fractional entangling transform, using Eq. (16), we can put the transform corresponding to the kernel [Eq. (10)] as $$\begin{gathered} Fθ[f](η′)=∫−∞∞d2ηπKC(η′,η)f(η) \\ =∫−∞∞d2ηπ〈η′|Uθ|η〉〈η||f〉 \\ =〈η′|exp{iθ(ab†+a†b)}|f〉, \end{gathered}$$(17)which is just the quantum version of the FrET. This indicates that the FrET just corresponds to the matrix element of unitary operator exp{iθ(ab†+a†b)} in the entangled state representation 〈η′| and the signal vector |f〉.

Using the quantum version of FrFT and the completeness relation of the entangled state [Eq. (7)], it is easy to see that $$\begin{gathered} Fθ+α[f(η)]=〈η′|exp{i(θ+α)(ab†+a†b)}|f〉 \\ =∫−∞∞d2η′′π〈η′|exp{iθ(ab†+a†b)}|η′′〉 \\ ×∫−∞∞d2ηπ〈η′′|exp{iα(ab†+a†b)}|η〉f(η) \\ =∫−∞∞d2η′′π〈η′|exp{iθ(ab†+a†b)}|η′′〉Fα[f(η)] \\ =FθFα[f(η)]. \end{gathered}$$(18)

Thus the FrET actually satisfies the fractional additivity. The proof is clear and concise by using the representation 〈η′| and the quantum version of the FrET.

Regarding applications, the quantum version [Eq. (17)] of the FrET can help us to derive the FrET of some wave functions conveniently. In this Letter, we consider the two-mode number state as the signal vector, |f〉=|m,n〉=a†mb†n/m!n!|00〉. Noting the coherent state representation of the number state |m〉=1/m!(∂m/∂αm)|α〉|α=0, where |α〉=exp{αa†}|0〉 is the unnormalized coherent (〈0|α〉=1), thus $$\begin{gathered} f(η)=e−12|η|2m!n!∂m+n∂αm∂βnexp[η*α−ηβ+αβ]|α,β=0 \\ =im+ne−12|η|2m!n!Hm,n(−iη*,iη), \end{gathered}$$(19)where Hm,n(x,y) are two-variable Hermite polynomials whose mother function is defined by^[15]Hm,n(x,y)=∂m+n∂tm∂τnexp[−tτ+tx+τy]|t=τ=0.(20)

On the other hand, note the transform relation Uθa†Uθ†=a†cosθ+ib†sinθ, and Uθb†Uθ†=b†cosθ+ia†sinθ, it is easy to get $$\begin{gathered} Uθ|m,n〉=1m!n!(a†cosθ+ib†sinθ)m(b†cosθ+ia†sinθ)n|00〉 \\ =1m!n!∂m+n∂αm∂βnea†(αcosθ+iβsinθ)+b†(iαsinθ+βcosθ)|00〉|α,β=0 \\ =1m!n!∂m+n∂αm∂βn|α¯,β¯〉|α,β=0, \end{gathered}$$(21)where |α¯,β¯〉 (α¯=αcosθ+iβsinθ, β¯=iαsinθ+βcosθ) are also unnormalized coherent states. Thus, using Eqs. (17) and (19) the FrET of the Hermite Gaussian function [Eq. (19)] can be calculated as $$\begin{gathered} Fθ[f(η)]=〈η′|Uθ|m,n〉 \\ =e−12|η′|2m!n!∂m+n∂αm∂βnexp[η′*α¯−η′β¯+α¯β¯]|α,β=0 \\ =∑l=0min(m,n)(iisin2θ2)m+n−2ln!m!e−12|η′|2cosl2θ4ll!(n−l)!(m−l)! \\ ×Hm−l(η¯i2isin2θ)Hn−l(η¯*i2isin2θ), \end{gathered}$$(22)where η¯=η′*cosθ−iη′sinθ and we have used the mother function formula of single-variable Hermite polynomials $$\begin{gathered} Hn(x)=∂n∂tnexp(2xt−t2)|t=0, \\ ddxlHn(x)=2ln!(n−l)!Hn−l(x). \end{gathered}$$(23)

Equation (22) is just the FrET of two-mode number state which is related to the single-variable Hermite polynomials.

In conclusion, based on the complex FrFT, we successively propose a new kind of fractional transform (the FrET) by replacing the integration kernel with a new one. Employing the IWOP technique and the entangled state representation, the unitary operator corresponding to the FrET is just a entangling operator and is evidently different from that in the complex FrFT. The additivity property, an important fractional feature, is easily proved by using the matrix element expression and the completeness relation of entangled state representation in the framework of quantum mechanics. We should point out that we not only find the quantum mechanical correspondence (unitary operator) of the classical optical transform, but also develop the classical one from the viewpoint of quantum optics. That is to say, it is possible that some other transforms can be presented by using different quantum mechanical unitary operators or representations. In our derivation, the IWOP technique plays an important role.