YesTwo fractional Fourier transforms are used to define bi-fractional displacement operators, which interpolate between displacement operators and parity operators. They are used to define bi-fractional coherent states. They are also used to define the bi-fractional Wigner function, which is a two-parameter family of functions that interpolates between the Wigner function and the Weyl function. Links to the extended phase space formalism are also discussed

A Vourdas

C Lei

Chong C C

Chountasis S

Ozaktas H M

S Agyo

Zachos C K

Journal of Physics : Conference Series

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Journal of Physics Conference Series

Bi-fractional Wigner functions

Agyo, Sanfo D.

Lei, Ci

Vourdas, Apostolos

Bradford Scholars

 The University of Bradford Institutional Repository http://bradscholars.brad.ac.uk This work is made available online in accordance with publisher policies. Please refer to the repository record for this item and our Policy Document available from the repository home page for further information. To see the final version of this work please visit the publisher’s website. Access to the published online version may require a subscription. Link to publisher version: http://dx.doi.org/10.1088/1742-6596/597/1/012007  Citation: Agyo S, Lei C and Vourdas A (2016) Bi-fractional Wigner functions. Journal of Physics: Conference series. 597. Conference 1. Copyright statement: © 2016 The Authors. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.   Bi-fractional Wigner functionsS. Agyo, C. Lei, A. VourdasDepartment of Computing,University of Bradford,Bradford BD7 1DP, United KingdomE-mail: C.Lei1@bradford.ac.ukAbstract. Two fractional Fourier transforms are used to define bi-fractional displacementoperators, which interpolate between displacement operators and parity operators. They areused to define bi-fractional coherent states. They are also used to define the bi-fractional Wignerfunction, which is a two-parameter family of functions that interpolates between the Wignerfunction and the Weyl function. Links to the extended phase space formalism are also discussed.1. IntroductionWigner and Weyl functions and also P and Q functions play a central role in the generalarea of phase space methods[1, 2]. They are quantities which describe pseudo-probabilities andcorrelations associated to quantum states. There have also been other more general quantities,which interpolate among them, and provide a more general formalism. In this general contextin a recent publication [3], we presented a two-parameter interpolation between displacementand parity operators (or similarly between Weyl and Wigner functions). The displacement andparity operators are related through a two-dimensional Fourier transform, which we replace withfractional Fourier transforms [4, 5, 6, 7]. This leads to a two-parameter family of operators whichwe call bi-fractional displacement operators, and which include as special cases the displacementand parity operators.In the present paper we review briefly and extend this work, by discussing links with theextended phase space method in refs[8, 9, 10, 11, 12]. The latter uses the fact that the Wignerfunction W (x, p) is related to the Weyl function W˜ (X,P ) through a two-dimensional Fouriertransform. x, p are position and momentum entering in the Wigner function, and X,P areposition and momentum increments entering in the Weyl function. One Fourier transforminvolves the variables x, P and the other the p,X. These Fourier transforms lead to uncertaintyrelations between the x, P variables (and also the X, p variables), which have been studied in[13]. An ‘extended Wigner function’ (and other ‘extended quantities’) that depend on x, p,X, Phave been introduced in the extended phase space x − p − X − P . In this language, the twovariables of the bi-fractional displacement operator studied here, are in the direction defined bythe angle φ1 in the x− P plane, and also in the direction defined by the angle φ2 in the p −Xplane (see fig1). Using the bi-fractional displacement operator we get a two-parameter familyof functions which include as special cases the Wigner and Weyl functions.In section 2, we discuss briefly fractional Fourier transforms, in order to define the notation.In section 3, we introduce the bifractional displacement operators. In section 4, we act with themon the vacuum to get bi-fractional coherent states. In section 5, we introduce the ‘bi-fractionalWigner functions’, which is a two parameter family of functions, that includes Wigner and Weylfunctions as special cases. In section 6, we link the present work with the extended phase spaceformalism. We conclude in section 7, with a discussion of our results.2. Fractional Fourier TransformThe fractional Fourier Transform has been studied extensively for a long time. It is ageneralisation of the Fourier transform given byfφ(α) = exp(iφA†A)f(α); A = α+ ∂α√2; A† = α− ∂α√2(1)This is the differential form and it is equivalent to the following integral formfφ(α) =∫Kφ(α, β)f(β)dβ, (2)where Kφ(α, β) is the kernelKφ(α, β) =[1+i cot φ2pi]exp[−i(α2+β2) cot φ2 +iαβsinφ], if φ 6= npiδ(α − β), if φ = 2npiδ(α + β), if φ = (2n + 1)pi(3)It is easily seen that for φ = pi/2 we get the Fourier transform, and for φ = pi the parity operatorwith respect to the origin which we denote as P(0, 0) . We note that,∫dβ Kφ1(α, β)Kφ2(β, γ) = Kφ1+φ2(α, γ) (4)The two-dimensional fractional Fourier transform is given byfφ1,φ2(α, β) =∫Kφ1(β, α′)Kφ2(α,−β′)f(α′, β′)dα′dβ′ (5)3. Bi-fractional Displacement operatorWe consider the harmonic oscillator in the Hilbert space, H with xˆ and pˆ as the position andmomentum operators. We define the displacement operator asD(X,P ) = exp(i√2Pxˆ− i√2Xpˆ). (6)The bi-fractional displacement operator is defined asBφ1,φ2(α, β) = [cos(φ1 − φ2)]1/2∫Kφ2(β,X)Kφ1(α,−P )D(X,P )dX dP. (7)Bφ1,φ2(α, β) is a continuous function of φ1, φ2. There is an important difference betweenEqs(5),(7). In Eq.(5) α′, β′ are independent variables. In Eq.(7) X,P are dual quantumvariables, multiplied by xˆ, pˆ which do not commute. Integrations should be performed verycarefully taking into account the ordering of the operators xˆ, pˆ. A result of this is the prefactor[cos(φ1 − φ2)]1/2 which is important for unitarity.In [3] we have proved that the Bφ1,φ2(α, β) are elements of the group HW ⋊SU(1,1) whichis the semidirect product of the Heisenberg Weyl group HW of displacements, by the SU(1,1)group of squeezing transformations. They are ‘special elements’ in the sense that not everyelement of this group can be written as Bφ1,φ2(α, β).A special case of Bφ1,φ2(α, β) is the parity operator with respect to the point (α, β), which isdefined asP(α, β) = D(α2,β2)P(0, 0)D†(α2,β2)= D(α, β)P(0, 0) (8)The parity operator is related to the displacement operator through a two-dimensional FouriertransformP(x, p) = 12pi∫D(X,P ) exp [i(pX − Px)] dX dP=∫D(X,P ) Kpi2(p,X) Kpi2(x,−P )dX dP (9)For φ1 = φ2 =pi2 , we getBpi2,pi2(α, β) = P(α, β) (10)The bi-fractional displacement operator Bφ1,φ2(α, β) is unitary:[Bφ1,φ2(α, β)]† = B−φ1,−φ2(−α,−β) (11)Special cases are:B0,0(αβ) = D(β,−α)Bpi2,pi2(α, β) = P(α, β)Bpi,pi(α, β) = D(−β, α) (12)The relationship between two bi-fractional displacement operators with different variables isBψ1+φ1,ψ2+φ2(α, β) =| cos(ψ1 + φ1 − ψ2 − φ2)|1/2| cos(ψ1 − ψ2)|1/2×∫dσdωKφ2(β, β′)Kφ1(α,α′)Bψ1,ψ2(α′, β′) (13)Acting with the displacement operator on both sides of the bifractional displacement operatorwe getD(ν, ξ)Bφ1,φ2(α, β)D†(ν, ξ) = Bφ1,φ2(−α− 2ν sinφ1, β + 2ξ sinφ2)× exp [i(2αν cosφ1 + ν2 sin 2φ1)]× exp [i(2βξ cosφ2 + ξ2 sin 2φ2)] (14)4. Bifractional coherent statesActing with Bφ1,φ2(α, β) on the vacuum |0〉 we get the generalized coherent states|α, β;φ1, φ2〉 = Bφ1,φ2(α, β) |0〉 (15)In the special case when φ1 = φ2 = 0, and φ1 = φ2 =pi2 we get Glauber coherent states:|−β, α; 0, 0〉 = |α, β; pi2,pi2〉 = |α, β〉 (16)They are eigenstates of the creation and annihilation operatorsb(φ1, φ2) = Bφ1,φ2(0, 0)a Bφ1,φ2(0, 0)b†(φ1, φ2) = Bφ1,φ2(0, 0)a†Bφ1,φ2(0, 0), (17)but they have novel non-trivial properties with respect to a, a†. They also satisfy the resolutionof the identity12pi∫|α, β;φ1, φ2〉 〈α, β;φ1, φ2| dαdβ = 1 (18)In [3] we have studied the Bargmann functions of these coherent states. Also the whole formalismof coherent state quantization, can be studied using the bifractional coherent states.5. Bi-fractional Wigner functionsWe consider the trace of a trace-class operator Θ with Bφ1,φ2(α, β):Vφ1,φ2(α, β|Θ) = Tr[ΘBφ1,φ2(α, β)]= | cos(φ1 − φ2)|1/2∫Kφ2(β,X)× Kφ1(α,−P )W˜(X,P |Θ)dXdP (19)This is a generalization of the Wigner and Weyl functions. IndeedV0,0(α, β|Θ) = W˜(β,−α|Θ)Vpi2,pi2(α, β|Θ) =W(α, β|Θ)Vpi,pi(α, β|Θ) = W˜(−β, α|Θ) (20)In the special case that φ1 = φ2 = 0, the variable α becomes the variable P , the variableβ becomes the variable X (see fig.1), and Vφ1,φ2(α, β|Θ) is the Weyl function. In the specialcase that φ1 = φ2 =pi2 , the variable α becomes the variable x, the variable β becomes thevariable p, and Vφ1,φ2(α, β|Θ) is the Wigner function. In general, the function Vφ1,φ2(α, β|Θ) isa two-parameter interpolation between the Wigner and Weyl functions.When Θ = ρ is a density matrix, Vφ1,φ2(α, β|ρ) is the bi-fractional Wigner function for thestate described by ρ. As example, we consider the superposition of coherent states|s〉 = N [|α0, β0〉+ |−α0,−β0〉] (21)where N is a normalization factor. In figs 2, 3 we plot ℜ[Vpi4,pi4(α, β|Θ)] and |Vpi4,pi4(α, β|Θ)| forthis state with α0 = 1.8 and β0 = 0. In the Weyl function the auto-terms are in the middle andthe cross-terms are in the ‘wings’, while the opposite is true in the Wigner function. Here wehave both auto-terms and cross-terms in the wings.6. Links with the extended phase space formalismThe extended phase space formalism uses the fact that the Wigner function W (x, p) and theWeyl function W˜ (X,P ) are related through a two-dimensional Fourier transform. The Wignerfunction describes quantum noise, and the Weyl function quantum correlations. Starting froma function g(u) with u ∈ RN , and its Fourier tranform g˜(v) with v ∈ RN , we can introduce(in any context) ‘a Wigner function’ W (u, v) (and other quantities) in the u − v phase space,which is R2N . The extended phase space formalism introduces extended Wigner functions thatFigure 1. The α direction in the x− P plane within the extended phase space x− p−X − P .Also the β direction in the p −X plane. For clarity, the two planes are drawn separately, butthey belong to the same four-dimensional space.) )α βφ1 φ2XPx pFigure 2. ℜ[Vpi4,pi4(α, β|Θ)] for the state of Eq.(21), with α0 = 1.8 and β0 = 0.depend on four variables (x, p,X, P ) and studies transformations in the extended phase spacex−p−X−P . The x−P plane links quantum noise in the x-direction, with quantum correlationsin the P -direction. Uncertainty relations in the x − P plane have been studied in [13]. Theyare related to the usual uncertainties in the case of pure states, but they are different for mixedstates. Analogous comments can be made for the p−X plane.In the extended phase space x − p − X − P we can introduce the following dispacementoperator of Weyl functions:D(A1, A2, B1, B2) = exp[(B2 −A2)∂P ] exp(−iA1 +B12P)exp[(B1 −A1)∂X ]× exp(iA2 +B22X)(22)Figure 3. |Vpi4,pi4(α, β|Θ)| for the state of Eq.(21), with α0 = 1.8 and β0 = 0.Acting with this on the Weyl function W˜ (X,P |Θ) we get[9]D(A1, A2, B1, B2)W˜ (X,P |Θ) = exp(−iA1 +B12P + iA2 +B22X − iA1B2 + iA2B1)× W˜ (X +B1 −A1, P −A2 +B2) (23)We call Θ′ the operator with this Weyl function:W˜(X,P |Θ′) = D(A1, A2, B1, B2)W˜(X,P |Θ) (24)In this language, the present paper considers quantities in the general α − β plane (fig.1),which becomes the X − P plane in the special case φ1 = φ2 = 0, and the x − p plane in thespecial case φ1 = φ2 =pi2 . It then studies relations between the various quantities. For example,we show that Vφ1,φ2(α, β|Θ′) is related to W˜(X,P |Θ) as follows:Vφ1,φ2(α, β|Θ′) = | cos(φ1 − φ2)|1/2∫Kφ2(β,X)Kφ1(α,−P )× [D(A1, A2, B1, B2)W˜(X,P |Θ)]dXdP= | cos(φ1 − φ2)|1/2∫Kφ2(β,X + µ2) exp (iν2X)× Kφ1(α,−P − µ1) exp (−iν1P ) W˜(X,P |Θ)dXdP. (25)Hereµ2 = A1 −B1; ν2 = A2 +B22; µ1 = A2 −B2; ν1 = A1 +B12. (26)We next introduce the displaced kernel of the fractional Fourier transformKφ(x, y|κ, λ) = exp(iκy) exp(λ∂y)Kφ(x, y) = exp(iκy)Kφ(x, y + λ), (27)and rewrite Eq.(25) in the compact formVφ1,φ2(α, β|Θ′) = | cos(φ1 − φ2)|1/2∫Kφ2(β,X|ν2, µ2)Kφ1(α,−P | − ν1,−µ1)× W˜(X,P |Θ)dXdP (28)Other similar relations can also be proved. In particular symplectic Sp(4, R) transformations inthe x− p−X − P extended phase space require further study.7. DiscussionWe have studied the bi-fractional displacement operators Bφ1,φ2(α, β). They include as specialcases, both the displacement operators and the parity operators (Eq.(12)). We stress theimportance of the prefactor [cos(φ1 − φ2)]1/2 in Eq.(7), for unitarity. This is related to thefact that in Eq.(7) X,P are dual quantum variables, multiplied by xˆ, pˆ which do not commute.Using them we have defined the bi-fractional coherent states in Eq.(15). We have also definedthe bi-fractional Wigner functions in Eq.(19), which include as special cases, both the Wignerand Weyl functions (Eq.(20)).We have interpreted the formalism within the extended phase space, which relates quantumnoise with correlations. The bi-fractional displacement operators Bφ1,φ2(α, β), are displacementoperators in the α− β plane of fig.1. The formalism provides an interpolation between variousquantities in phase space.For further work we suggest the application of these ideas to finite quantum systems withvariables in Zd (the integers modulo d). We expect this generalization to be easier for prime d,in which case Zd is a field. Overall, the merit of the formalism is that different quantities appearto be special cases of a single quantity.References[1] Zachos C K, Fairlie D B and Curtright T L 2005 Quantum Mechanics in Phase Space (Singapore: WorldScientific)[2] Schleich W P 2001 Quantum Optics in Phase Space (New York: Wiley)[3] Agyo S, Lei C and Vourdas A 2015 Phys. Lett. A 379 255[4] Namias V 1980 J. Inst. Math. Applic. 25 241[5] McBride A C and Kerr F H 1987 IMA J. Appl. Math 39 159[6] Bailey D H and Swarztrauber P N 1991 SIAM Review 33 389[7] Ozaktas H M, Zalevsky Z and Kutay M A 2001 The Fractional Fourier Transform (New York: Wiley)[8] Chountasis S and Vourdas A 1998 Phys. Rev. A 58 1794[9] Chong C C and Vourdas A 2001 J. Phys. A 34 9849[10] Chountasis S and Vourdas A 1999 J. Phys. A 32 6949[11] Ponomarenko S A and Wolf E 2001 Phys. Rev. A 63 062106[12] Agarwal G S and Ponomarenko S A 2003 Phys. Rev. A 67 032103[13] Vourdas A 2004 Phys. Rev. A 69 022108

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Bi-fractional Wigner functions

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