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PHYSICAL REVIEW A NOVEMBER 1999VOLUME 60, NUMBER 5
Fractional Fourier operators and generalized Wigner functions
S. Chountasis,1 A. Vourdas,1 and C. Bendjaballah21Department of Electrical Engineering and Electronics, The University of Liverpool, Brownlow Hill,
Liverpool L69 3BX, United Kingdom2Laboratoire des Signaux et Systemes, Ecole Superieure d’ Electricite, Plateau de Moulon, 91192, Gif-Sur-Yvette Cedex, Fr
~Received 10 February 1999!
Fractional Fourier operators are introduced, and their properties are studied. Products of these operators withthe displacement operators are also considered and used to define generalized Wigner functions which inspecial cases give the known Wigner functions and Weyl functions. The properties of these generalized Wignerfunctions are explored. Complex fractional Fourier operators are also studied.@S1050-2947~99!01411-0#
PACS number~s!: 03.65.Bz, 42.50.Dv
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I. INTRODUCTION
Fourier transforms play an important role in variobranches of science and engineering. In quantum mechathey relate dual representations, e.g., the position represtion s(x) with the momentum representations(p). Recentlythere has been a lot of work on Fourier transform incontext of quantum systems~e.g., Refs.@1,2#!. This workmight lead to novel quantum devices that perform Foutransforms~eg., ~Ref. @3#!.
An interesting generalization has been the concept of ftional Fourier transform which appeared mainly in a maematical and classical optical signal processing context@4,5#.In this paper we extend previous work on fractional Fourtransform in a quantum context. We introduce the fractioFourier operatorV(u) which in the phase spacex-p rotatesthe wave functions(x) into the wave functions(x8), wherex8 is an axis at an angleu with thex axis. In Sec. II we studythe properties of the fractional Fourier operator, and presnumerical results which elucidate their meaning. In Sec.we introduce products of the fractional Fourier operatwith the displacement operators, and show that they lnaturally to a generalized Wigner function which foru5p isthe usual Wigner function and foru50 is the Weyl function.The properties of this generalized Wigner function are dcussed and exemplified with numerical examples. In Secwe consideru to be a complex number in which caseV(u) isno longer a unitary operator. There is merit in studying toperator, for applications with losses, and we exploreproperties. We conclude in Sec. V with a discussion ofresults.
II. FRACTIONAL FOURIER OPERATORS
We consider the harmonic oscillator Hilbert spaceHspanned by the number eigenstates$uN&;N50,1,2 . . .% anduse units where\5kB5c51. We also consider the coherestates
uA&5expS 2uAu2
2 D (N50
`AN
~N! !1/2uN&5D~A!u0&, ~1!
D~A!5exp@Aa†2A* a#, ~2!
PRA 601050-2947/99/60~5!/3467~7!/$15.00
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where a† and a are the usual creation and annihilation oerators.D(A) is the displacement operator which can alsoexpressed in terms of the position and momentum operax and p as
D~x,p
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3468 PRA 60S. CHOUNTASIS, A. VOURDAS, AND C. BENDJABALLAH
F[VS p
2 D5expS ip
2a†aD5 (
N50
`
i NuN&^Nu. ~13!
Indeed, it is easily seen that
F451, ~14!
Fux&5up&, Fup&5u2x&, ~15!
FxF†5 p, FpF†52 x. ~16!
For later purposes we introduce the eigenstates of the optors xu andpu :
ux;u&[V~u!ux&, up;u&[V~u!up&. ~17!
It is easily seen thatux;p/2&5up&, ux;p&5u2x&, ux;3p/2&5u2p&. We can prove that
D~x,y;u
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PRA 60 3469FRACTIONAL FOURIER OPERATORS AND . . .
U~x1 ,p1 ;u1!U~x2 ,p2 ;u2!5U~x8,p8;u11u2!eig,~28!
where
x85x11x2 cosu12p2 sinu1 ,
p85p11x2 sinu11p2 cosu1 , ~29!
g52~x1p21x2p1! cosu122~x1x22p1p2! sinu1 .
The Wigner function@6–8# of a state described by a density matrix r can be expressed in terms of the displacparity operator@9# U(x,p;p)
W~x,p!5Tr@rU~x,p;p!#
5 12 E dX^x1 1
2 Xurux2 12 X&exp~2 iXp!
5 12 E dP^p1 1
2 Purup2 12 P&exp~ iPx!. ~30!
The equivalence between these expressions is known@6–10#.Another function which is useful in phase-space methodthe Weyl function@11# which can be expressed in termsthe operatorU(X,P;0) as
W~X,P!5Tr@rU~X,P;0!#
5E dx^x1Xurux2X&exp~2 i2Px!
5E dp^p1Purup2P&exp~ i2pX!. ~31!
FIG. 2. Fractional Fourier transform of the wave functions(x)for the superposition of two coherent states of Eq.~25! with A52.8 and foru5p/3. The solid line represents the real part, anddotted line the imaginary part.
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is
The equivalence between these two expressions is kn@11#. The Wigner function is related to the Weyl functiothrough the Fourier transform@6–11#
W~X,P!51
pE E dx dp W~x,p!exp@2 i2~Px2pX!#.
~32!
It is seen that the Wigner and Weyl functions are definin terms of the operatorsU(x,p;p) and U(X,P;0) corre-spondingly. In this paper we use the general operaU(x,p;u) to define theWu function as
Wu~x,p!5Tr@rU~x,p;u!#
5E E dy dz yuruz12x&
3exp@ i2p~z1x!#D~y,z;u!. ~33!
It was proved in Ref.@10# that
1
2pE dx D~x,p!5U12 pL K 21
2pU, ~34!
1
2pE dp D~x,p!5U12 xL K 21
2xU, ~35!
1
4pE dx dp D~x,p!5V~p!. ~36!
Multiplying these by the operatorV(u) from the right, weprove
1
pE dx U~x,p;u!5up&^p;p2uu, ~37!
1
pE dp U~x,p;u!5ux&^x;p2uu, ~38!
1
pE dx dp U~x,p;u!5V~u2p!. ~39!
Using these we obtain:
1
pE dx Wu~x,p!5^p;p2uurup&, ~40!
1
pE dp Wu~x,p!5^x;p2uurux&, ~41!
1
pE dx dp Wu~x,p!5Tr@rV~u2p!#. ~42!
We have numerically evaluated the generalized Wigfunction Wu(x,p) for the superposition of two coherenstates of Eq.~25! with A52.8. In Fig. 3 we present resultfor u50, in which caseW0(x,p) is the Weyl function whichin this particular example is real. The exact values dependthe normalization. The results presented here are basethe normalization shown in Eq.~25! in conjuction with thedefinition of Eq. ~33!. Results foru5p/4 are presented in
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3470 PRA 60S. CHOUNTASIS, A. VOURDAS, AND C. BENDJABALLAH
Figs. 4 and 5. Here we present the intersectionsWp/4(x,0)andWp/4(0,p) correspondingly, so that the exact values cbe clearly seen. Results foru5p/2 are presented in Fig. 6~real part! and Fig. 7~absolute value!. In Fig. 8 we presentresults foru5p, in which caseWp(x,p) is the Wigner func-tion. Note that in the Wigner function (u5p) shown in Fig.8, the two Gaussians represent autoterms, and the oscillpart in the middle represents cross-terms. In the Weyl fution ~Fig. 3! it is the other way around: the oscillating paaround the origin represents autoterms and the two Gaians represent cross-terms~for a discussion see Ref.@11#!. Itis interesting to see that asu decreases fromp to 0 theautoterms, which are originally (u5p) Gaussians, becomoscillatory and they move in the phase plane endingu50) at the origin. The cross-terms, which are originallyu5p) oscillatory, eventually become (u50) Gaussians.
FIG. 3. The functionWu(x,p) for the superposition of two coherent states of Eq.~25! with A52.8 and foru50. In this caseWu(x,p) is the Weyl function. In this particular example, the imagnary part ofWu(x,p) is zero.
FIG. 4. The real~solid line! and imaginary~dotted line! parts ofthe functionWu(x,0) for the superposition of two coherent statesEq. ~25! with A52.8 and foru5p/4.
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The quantum tomography method@12# of reconstructingthe Wigner function from optical homodyne measuremecan also give the generalized Wigner functionWu(x,p). Thequantity measured is the Radon transform of the Wigfunction along the linex sinu2pcosu5q, defined as
Q~q,u!5E E W~x,p!d~x sinu2p cosu2q!dx dp
5E W~q sinu1u cosu,2q cosu1u sinu!du,
~43!
whereq is a real variable and 0<u,p. From theQ(q,u)we can evaluate the Weyl function using the Fourier traform
W~X,P!5E QFq, tan21S P
XD Gexp@2 iq~X21P2!1/2#dq.
~44!
f
FIG. 5. The real~solid line! and imaginary~dotted line! parts ofthe functionWu(0,p) for the superposition of two coherent statesEq. ~25! with A52.8 and foru5p/4.
FIG. 6. The real part of the functionWu(x,p) for the superpo-sition of two coherent states of Eq.~25! with A52.8 and foru5p/2.
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PRA 60 3471FRACTIONAL FOURIER OPERATORS AND . . .
This formula can be proved easily by taking the Fouritransform of both sides in Eq.~43!. We then insert the rela-tion @6,7,10#
r52
pE E da db D†~2a,2b!Tr@rD~2a,2b!#
52
pE E da db D†~2a,2b!W~a,b! ~45!
into Eq. ~33!, and obtain
Wu~x,p!52
pE E da db W~a,b!
3Tr@D†~2a,2b!D~2x,2p!V~u!#; ~46!
since
D†~2a,2b!D~2x,2p!5D~2x22a,2p22b!
3exp@ i ~2ap22bx!#, ~47!
we obtain
FIG. 7. The absolute value of the functionWu(x,p) for thesuperposition of two coherent states of Eq.~25! with A52.8 and foru5p/2.
FIG. 8. The functionWu(x,p) for the superposition of two co-herent states of Eq.~25! with A52.8 and foru5p. In this caseWu(x,p) is the Wigner function.
r
Wu~x,p!52
pE E da dbW~a,b!
3Tr@U~x2a,p2b;u!#exp@ i ~2ap22bx!#.
~48!
The trace of the operatorU(x,p;u1 i e) can be found to be
Tr@U~x,p;u1 i e!#5 (N50
`
exp@N~ iu2e!#^NuD~2x,2p!uN&
5exp@2~x21p2!# (N50
`
3exp@N~ iu2e!#LN~2x212p2!
51
12exp~ iu2e!
3expF ~x21p2!exp~ iu2e!11
exp~ iu2e!21G . ~49!
The result for^NuD(2x,2p)uN& was given in Ref.@13# interms of the Laguerre polynomialsLN . Evaluation of thesum was given in Ref.@14# and it exists fore.0, but di-verges fore,0. In the casee50, which is of interest to ushere, it exists as a one-sided limit. Combining Eqs.~48!,~49!, and~44!, we obtain
Wu~x,p!52
pE E E da db dq QFq,tan21S b
a D G3exp@2 iq~a21b2!1/2#exp@ i ~2ap22bx!#
3expF ~x2a!21~p2b!2
i tan~u/2! G 1
12exp~ iu!. ~50!
Using this equation we can reconstruct the generaliWigner functionWu(x,p) from quantum tomography measurements.
In the special caseu5p, Eq. ~50! reduces to
Wp~x,p!51
pE E E da db dq QFq,tan21S b
a D G3exp@2 iq~a21b2!1/2#exp@ i ~2ap22bx!#.
~51!
Taking into account Eq.~44! and the inverse of the Fourietransform of Eq.~32!, we see thatWp(x,p) is indeed theWigner function, as it should be.
IV. COMPLEX FRACTIONAL FOURIER OPERATORS
In this section the variableu is a complex number,u5uR1 iu I . Physically, complexu with positiveu I describescases with attenuation~‘‘filtering’’ !, and, with negativeu I ,cases with amplification. Equation~18! is still valid, pro-vided thatu I>0. For negativeu I the sum of Eq.~18! di-verges~due to continuous amplification!. Here we considerthe caseu I>0. V(u) is no longer a unitary operator, and cabe written as
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3472 PRA 60S. CHOUNTASIS, A. VOURDAS, AND C. BENDJABALLAH
V~u!5exp~2u I a†a!V~uR!, V~uR!5exp~ iuRa†a!,
~52!
where V(uR) is a unitary operator. Defining xuR
5V(uR) xV(uR)† and puR5V(uR) pV(uR)†, we can general-
ize Eqs.~7! and ~8! to
xu[V~u!xV~u!†5exp~2u I a†a!xuR
exp~2u I a†a!
5exp~22u I a†a!@ xuR
cosh~u I !2 i puRsinh~u I !#,
~53!
pu[V~u! pV~u!†5exp~2u I a†a!puR
exp~2u I a†a!
5exp~22u I a†a!@ i xuR
sinh~u I !1 puRcosh~u I !#.
~54!
Equation~18! is valid ~for u I>0), and therefore Eq.~23! isstill valid but the inverse transform given by Eq.~24! is nolonger valid. Using Eq.~23!, we have evaluateds(x,u) forthe superposition of two coherent states described in~25!. In Fig. 9 we present both the real~solid line! and imagi-nary part ~broken line! of s@x,(p/2)1 i #. In Fig. 10 wepresent both the real~solid line! and imaginary~broken line!parts ofs@x,(p/3)1 i #. It is seen that the values of the wav
FIG. 9. Fractional Fourier transform of the wave functions(x)for the superposition of two coherent states of Eq.~25! with A52.8 and foru5(p/2)1 i . The solid line represents the real paand the dotted line the imaginary part.
q.
functions here are much smaller than in the case of reau.This is related to the attenuation which the complexu de-scribes.
V. DISCUSSION
Motivated by recent work on fractional Fourier transformwe have introduced the fractional Fourier operator of Eq.~6!.Acting with this operator on a wave functions(x), we obtaina wave functions(x,u) which is another representatioalong the axisxu in the phase spacex-p, which forms anangleu with the axisx. We have also introduced productsfractional Fourier operators with displacement operatorsEq. ~26! and studied their properties. Taking the tracethese operators with the density matrix of a state, we oba generalized Wigner function, which in the special caseu5p reduces to the usual Wigner function, and in the specaseu50 reduces to the Weyl function. The propertiesthe generalized Wigner functions have been studied@Eqs.~37!–~42!#. We have also shown in Eq.~50! how they can beconstructed from quantum tomography measurements.
The case of complexu has also been considered, in reltion with physical problems with attenuation. Several eamples have been presented, but further work is requirethis interesting direction.
The work should be seen in the general context of Wigtomography@12#. We have demonstrated in this paper thWigner and Weyl functions are only special cases ofgeneral wave function of Eq.~33!, which could play an im-portant role in phase-space studies.
ACKNOWLEDGMENTS
One of us~S.C.! gratefully acknowledges financial support from the Alexander S. Onassis Public Benefit Fountion.
FIG. 10. Fractional Fourier transform of the wave functions(x)for the superposition of two coherent states foru5(p/3)1 i . Thesolid line represents the real part, and the dotted line the imaginpart.
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.
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PRA 60 3473FRACTIONAL FOURIER OPERATORS AND . . .
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