Collins diffraction formula and the Wigner function in entangled state representation

0.11 MB·English

by  Hong-yi Fan

Collins diffraction formula and the Wigner function in
entangled state representation
9
0 Hong-Yi Fan1,2 and Li-yun Hu1*
0
1
2 Department of Physics, Shanghai Jiao Tong University,Shanghai200030, China
2
n Department of Material Science and Engineering,
a
University of Scienceand Technology of China, Hefei, Anhui 230026, China
J
Corresponding author. E-mail address: [email protected].
4

January 4, 2009
]
h
p
-
t Abstract
n
a BasedonthecorrespondencebetweenCollinsdiffractionformula(opticalFresneltransform)
u
and the transformation matrix element of a three-parameters two-mode squeezing operator
q
in the entangled state representation (Opt. Lett. 31 (2006) 2622) we further explore the
[
relationship between output field intensity determined by the Collins formula and the input

1 field’sprobability distribution along an infinitelythin phasespacestrip both in spacial domain
v and frequency domain. The entangled Wigner function is introduced for recapitulating the
4 result.
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OCIS codes: 070.2590, 270.6570
3
0 In a preceding Letter [1] we have reported that the Collins diffraction formula in cylindrical
1. coordinates is just the transformation matrix element of a three-parameter (k and t are complex
0 and satisfy the unimodularity condition kk∗ tt∗ =1) two-mode squeezing operator [2, 3]
−
9
0 t t∗
v: F(t,k) =exp(cid:18)k∗a†1a†2(cid:19)exph(cid:16)a†1a1+a†2a2+1(cid:17)ln(k∗)−1iexp(cid:18)−k∗a1a2(cid:19), (1)
i
X in the deduced entangled state representation s,r′ ,
h |
r
a φ (r′) s,r′ φ = s,r′ F(t,k) ψ
s
≡ h | i h | | i
is ∞ i rr′
= d r2 exp Ar2+Dr′2 J ψ (r), (2)
2iB Z (cid:20)2B (cid:21) s(cid:18)− B (cid:19) s
0 (cid:0) (cid:1) (cid:0) (cid:1)
where ψ and φ denote the incoming and output light, respectively, [a ,a†]=δ ,
s s i j i,j
1 1
k = [A+D i(B C)], t= [A D+i(B+C)], (3)
2 − − 2 −
we see that the relation kk∗ tt∗ =1 becomes AD BC =1, J is the sth Bessel function, and
s
− −
1 2π
s,r′ = dθeisθ η =r′eiθ , (4)
h | 2π Z
0 (cid:10) (cid:12)
(cid:12)
here η is the entangled states in two-mode Fock space [4, 5, 6, 7] named after Einstein-Podolsky-
| i
Rosen (EPR)’s [8] concept of quantum entanglement,
1
η =exp η 2+ηa† η∗a†+a†a† 00 . (5)
| i {−2| | 1− 2 1 2}| i
1
Thus s,r′ F(t,k) ψ isthe quantumopticsversionoftheCollinsformula(generalizedHankeltrans-
h | | i
formation). In [9] we have also found
1 1 i
(t,k)(η′,η)= η′ F(t,k) η = exp A η 2 (ηη′∗+η∗η′)+D η′ 2 . (6)
K π h | | i 2iBπ (cid:26)2B h | | − | | i(cid:27)
Comparing with the integral kernel of usual Fresnel transform which describes how a general beam
ψ(x′),propagatingthroughan(ABCD) opticalparaxialsystem,becomes outputfieldφ(x)[10,11]
∞
φ(x)= (x,x′)ψ(x′)dx′, (7)
Z K
−∞
where AD BC =1,
−
1 i
(x,x′)= exp Ax′2 2x′x+Dx2 , (8)
K √2πiB (cid:20)2B − (cid:21)
(cid:0) (cid:1)
we see that (t,k)(η′,η) can be considered as the integration kernel of 2-dimensional entangled
K
optical Fresnel transform,
Ψ(η′)= (t,k)(η′,η)Φ(η)d2η, (9)
Z K
in this sense F(t,k) can be named entangled Fresnel operator (EFO), here Φ(η) = η Φ , Ψ(η′) =
η′ Ψ , and we have used the completeness relation d2η η η =1. h | i
h | i π | ih |
Clearly, if the [ABCD] system is changed to [D( RB)( C)A] system, then Eq. (9) should read
− −
Ψ(η′)= (D,−B,−C)(η′,η)Φ(η)d2η, (10)
Z K2
(D,−B,−C)
where is
K2
1 i
(D,−B,−C)(η′,η)= exp D η 2 (ηη′∗+η∗η′)+A η′ 2 . (11)
K2 2iBπ (cid:26) 2B h | | − | | i(cid:27)
− −
On the other hand, signals or images in optical information theory may be described directly or
indirectlybythe Wignerdistributionfunction(WDF) [12]. Inone-dimensional(1D)case,the WDF
of an optical signal field ψ(x) is defined as
+∞ du u u
W (ν,x)= eiνuψ∗ x+ ψ x . (12)
ψ
Z−∞ 2π (cid:16) 2(cid:17) (cid:16) − 2(cid:17)
W (ν,x)involvesbothspatialdistributioninformationandspace-frequencydistributioninformation
ψ
of the signal, ν is named space frequency. Now, let us consider the entangled case. Like Eq. (12),
it is natural to introduce the 2-D complex Wigner transform as
W(σ,γ)= d2ηψ(σ+η)ψ∗(σ η)eηγ∗−η∗γ, (13)
Z π3 −
where σ, γ, η are all complex variables. To see its physical meaning, using the integration formula
of Dirac δ function, we perform the following integration,
−
d2η 1
d2γW(σ,γ)= ψ(σ+η)ψ∗(σ η)δ(η)δ(η∗)= ψ(σ)2, (14)
Z Z π − π | |
which is just the probability distribution of the complex function ψ(σ). Further, let the ordinary
Fourier transforms of ψ(σ) be j(ζ),
ψ(σ)= d2ζj( ζ)e(ζ∗σ−ζσ∗)/2, (15)
Z 2π −
2
then substituting (15) into (13) leads to
W(σ,γ) = d2ηd2ζ d2ζ′j( ζ)j∗ ζ′ e(ζ∗−ζ/∗)σ2−(ζ−ζ/)σ∗eη„γ∗+ζ∗+2ζ/∗«−η∗„γ+ζ+2ζ′«
Z π3 2π 2π − (cid:16)− (cid:17)
= d2ζj( ζ)j∗(2γ+ζ)e(ζ∗+γ∗)σ−(ζ+γ)σ∗ = d2ζj(γ ζ)j∗(γ+ζ)eζ∗σ−ζσ∗(.16)
Z π3 − Z π3 −
It then follows from (16) that
d2σW (σ,γ) = d2ζj(γ ζ)j∗(ζ+γ) d2σeζ∗σ−ζσ∗
Z Z π3 − Z
d2ζ 1
= j(γ ζ)j∗(ζ+γ)δ(ζ)δ(ζ∗)= j(γ)2, (17)
Z π − π | |
whichis the probabilitydistribution ofthe complex function j(γ). Thus our definitionin (13) leads
to two marginal distributions in σ and γ phase space, respectively. Hence W (σ,γ) is indeed the
correctcomplex 2-D Wigner function (Wigner transform)of complex function ψ(σ) or j(γ). If one
wants to reconstruct the Wigner function by using various probability distributions, obviously the
“position density” σ ψ 2 and the space-frequency density γ ψ 2 are not enough, so we extend
|h | i| |h | i|
δ(η)δ(η∗) δ(η )δ(η ) to δ(η Dσ Bγ )δ(η Dσ +Bγ ) and generalize (14) to,
1 2 1 1 2 2 2 1
≡ − − −
R (η ,η ) π δ(η Dσ Bγ )δ(η Dσ +Bγ )W (σ,γ)d2σd2γ, (18)
2 1 2 1 1 2 2 2 1
≡ Z − − −
R (η ,η )isalsoaprobabilitydistributionalonganinfinitely thinphasespacestripdenotedbythe
2 1 2
real parameters B,D, which is a generalized entangled Radon transform [13, 14] of the two-mode
Wigner function (in the entangled form) [15, 16],
Thenaninterestingquestionnaturallyarises: whatistherelationbetweenthegeneralizedFresnel
transform and the WDF in entangled state representation?
We begin with rewriting the 2-D WF (13) as
W (σ,γ) = d2σ′d2σ′′ d2ηψ(σ′)ψ∗(σ′′)δ(2)(σ′ σ η)δ(2)(σ η σ′′)eηγ∗−η∗γ
Z Z π3 − − − −
= d2σ′d2σ′′ψ(σ′)ψ∗(σ′′)δ(2)(2σ σ′ σ′′)e(σ′−σ)γ∗−(σ′−σ)∗γ. (19)
Z π3 − −
Substituting (19) into (18) we rewrite the Radon transform of W (σ,γ) as (d2σ = dσ dσ , d2γ =
1 2
dγ dγ )
1 2
d2σ′d2σ′′
R (η ,η ) = ψ(σ′)ψ∗(σ′′) d2σd2γδ(η Dσ +Bγ )
2 1 2 Z π2 Z 2− 2 1
δ(η Dσ Bγ )δ(2)(2σ σ′ σ′′)e(σ′−σ)γ∗−(σ′−σ)∗γ
1 1 2
× − − − −
d2σ′d2σ′′ σ′ +σ′′
= ψ(σ′)ψ∗(σ′′) d2γδ η D 2 2 +Bγ
Z 4π2 Z (cid:18) 2− 2 1(cid:19)
σ′ +σ′′
δ η D 1 1 Bγ exp i[(σ′ σ′′)γ (σ′ σ′′)γ ]
× (cid:18) 1− 2 − 2(cid:19) { 2− 2 1− 1− 1 2 }
d2σ′d2σ′′
= ψ(σ′)ψ∗(σ′′)
Z 4B2π2
i σ′ +σ′′ σ′ +σ′′
exp (σ′ σ′′) η +D 2 2 (σ′ σ′′) η D 1 1
× (cid:26)B (cid:20) 2− 2 (cid:18)− 2 2 (cid:19)− 1− 1 (cid:18) 1− 2 (cid:19)(cid:21)(cid:27)
d2σ′d2σ′′ i
= ψ(σ′)ψ∗(σ′′)exp D σ′ 2 σ′′ 2 2η (σ′ σ′′) 2η (σ′ σ′′)(20).
Z 4B2π2 (cid:26)2B h (cid:16)| | −| | (cid:17)− 1 1− 1 − 2 2− 2 i(cid:27)
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On the other hand, when the beam Φ(η) propagates through the [D( B)( C)A] optical system,
− −
according to the Fresnel integration (10)-(11), we have
d2η d2η′′
Ψ(η′)2 = (D,−B,−C)(η′,η)Φ(η) ∗(D,−B,−C)(η′,η′′)Φ∗(η′′)
| | Z π K2 Z π K2
1 d2η i
= exp D η 2+(ηη′∗+η∗η′) A η′ 2 Φ(η)
4B2 Z π (cid:26)2B h− | | − | | i(cid:27)
d2η′′ i
exp D η′′ 2 (η′′∗η′+η′′η′∗)+A η′ 2 Φ∗(η′′)
×Z π (cid:26)2B h | | − | | i(cid:27)
1 d2η i
= Φ(η)Φ∗(η′′)exp D η′′ 2 η 2 2η′ (η′′ η ) 2η′ (η′′ η )(21),
4B2π2 Z π (cid:26)2B h (cid:16)| | −| | (cid:17)− 1 1 − 1 − 2 2 − 2 i(cid:27)
whichisthesameasR (η ,η )in(20).Socombining(20),(10)-(11)and(21)wereachtheconclusion
2 1 2
1 d2η i 2
exp D η 2 (ηη′∗+η∗η′)+A η′ 2 Φ(η)
(cid:12)(cid:12)−2iB Z π (cid:26)−2B h | | − | | i(cid:27) (cid:12)(cid:12)
(cid:12) (cid:12)
= π(cid:12) δ(η′ Dσ Bγ )δ(η′ Dσ +Bγ )W (σ,γ)d2σd2γ, (cid:12) (22)
Z 1− 1− 2 2− 2 1
where AD BC =1. The physicalmeaning of Eq. (22) is: when an input field propagatesthrough
−
an optical [D( B)( C)A] system, the energy density of the output field is equal to the Radon
− −
transform of the two-mode entangled Wigner function of the input field. So far as our knowledge
is concerned, this conclusion seems new. Eq. (22) is the relationship between the input amplitude
and output one in spatial-domain. Next we turn to the frequency domain.
If taking the matrix element of F(t,k) in the ξ representation which is conjugate to η , where
| i | i
theoverlap η ξ is η ξ = 1exp[(ξη∗ ξ∗η)/2],weobtainthe2-dimensionalGFTinits‘frequency
h | i h | i 2 −
domain’, i.e.,
1 d2ηd2η′
ξ′ F(t,k) ξ = ξ′ η′ η′ F(t,k) η η ξ
π h | | i Z π3 h | ih | | ih | i
1 d2ηd2η′ ξ′∗η′ ξ′η′∗+ξη∗ ξ∗η
= exp − − (t,k)(η′,η)
4Z π2 (cid:18) 2 (cid:19)K
1 i
= exp D ξ 2+A ξ′ 2 ξ′∗ξ ξ′ξ∗ N(ξ′,ξ), (23)
2i( C)π (cid:20)2( C)(cid:16) | | | | − − (cid:17)(cid:21)≡K2
− −
where the superscript N of N means that this transform kernel corresponds to the parameter
matrix N = [D, C, B,A].KT2hus if the [D, C, B,A] system is changed to N˜ = [A,C,B,D]
− − − −
system, the GFT in its ‘frequency domain’ is given by
Ψ(ξ′)= N˜ (ξ′,ξ)Φ(ξ)d2ξ, (24)
Z K2
where N˜(ξ′,ξ) is
K2
1 i
N (ξ′,ξ)= exp A ξ 2+D ξ′ 2 ξ′∗ξ ξ′ξ∗ . (25)
K2 2iCπ (cid:20)2C (cid:16) | | | | − − (cid:17)(cid:21)
It then follows from Eqs.(24) and (25) that
Ψ(ξ′)2 = N˜ (ξ′,ξ)Φ(ξ)d2ξ ∗N˜ (ξ′,ξ′′)Φ∗(ξ′′)d2ξ′′
| | Z K2 Z K2
1
= d2ξd2ξ′′Φ(ξ)Φ∗(ξ′′)
4π2C2 Z
i
exp A ξ 2 ξ′′ 2 +2ξ′ (ξ′′ ξ )+2ξ′ (ξ′′ ξ ) . (26)
× (cid:26)2C h (cid:16)| | −| | (cid:17) 1 1 − 1 2 2 − 2 i(cid:27)
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On the other hand, in similar to (18), we consider the integration transform,
R (ξ ,ξ )=π δ(ξ Aσ Cγ )δ(ξ Aσ +Cγ )W (σ,γ)d2σd2γ, (27)
2 1 2 1 1 2 2 2 1
Z − − −
R (ξ ,ξ ) is alsoaprobabilitydistributionalonganinfinitely thinphase spacestripdenotedby the
2 1 2
real parameters A,C. Substituting (19) into (27) yields
d2σ′d2σ′′
R (ξ ,ξ ) = ψ(σ′)ψ∗(σ′′) δ(2)(2σ σ′ σ′′)d2σd2γ
2 1 2 Z π2 Z − −
δ(ξ Aσ +Cγ )δ(ξ Aσ Cγ )e(σ′−σ)γ∗−(σ′−σ)∗γ
2 2 1 1 1 2
× − − −
d2σ′d2σ′′ σ′ +σ′′
= ψ(σ′)ψ∗(σ′′) d2γδ ξ A 2 2 +Cγ
Z 4π2 Z (cid:18) 2− 2 1(cid:19)
σ′ +σ′′ σ′ σ′′ σ′∗ σ′′∗
δ ξ A 1 1 Cγ exp − γ∗ − γ
1 2
× (cid:18) − 2 − (cid:19) (cid:20) 2 − 2 (cid:21)
d2σ′d2σ′′
= ψ(σ′)ψ∗(σ′′)
Z 4π2C2
i
exp A σ′ 2 σ′′ 2 +2ξ (σ′′ σ′)+2ξ (σ′′ σ′) , (28)
× (cid:26)2C h (cid:16)| | −| | (cid:17) 1 1 − 1 2 2 − 2 i(cid:27)
which is the same as Ψ(ξ′)2 in (26). So combining (28), (24)-(25), and (26) we can draw the
| |
conclusion
1 i 2
exp A ξ 2+D ξ′ 2 ξ′∗ξ ξ′ξ∗ Φ(ξ)d2ξ
(cid:12)(cid:12)2iCπ Z (cid:20)2C (cid:16) | | | | − − (cid:17)(cid:21) (cid:12)(cid:12)
(cid:12) (cid:12)
= π(cid:12) δ(ξ Aσ Cγ )δ(ξ Aσ +Cγ )W (σ,γ)d2σd2γ.(cid:12) (29)
1 1 2 2 2 1
Z − − −
This is the relationship between the output amplitude and input one’s entangled Wigner function
in ‘frequency domain’.
In sum, based on the correspondence between Collins diffraction formula (optical Fresnel trans-
form)andthetransformationmatrixelementofathree-parameterstwo-modesqueezingoperatorin
the entangledstate representation,we have exploredthe relationshipbetween output field intensity
determined by the Collins formula and the input field’s probability distribution along an infinitely
thin phase space strip. The entangled Wigner function is introduced for recapitulating the result.
Work supported by the National NaturalScience Foundation of China (GrantNos 10775097and
10874174).
References
[1] H.-Y. Fan and H.-L. Lu, “Collins diffraction formula studied in quantum optics,” Opt. Lett.
31, 2622-2624(2006).
[2] D. F. Walls, “Squeezed states of light,” Nature 324, 210 (1986).
[3] H.P.Yuen, “Two-photoncoherentstatesofthe radiationfield,” Phys.Rev.A13,2226(1976).
[4] H. Y. Fan, H. R. Zaidiand J. R. Klauder, “New approachfor calculating the normallyordered
form of squeeze operators,” Phys. Rev. D 35, 1831 (1987).
[5] H.Y.Fan,“OperatorordringinquantumopticstheoryandthedevelopmentofDirac’ssymbolic
method,” J. Opt. B: Quantum Semiclassical Opt. 5, R147 (2003).
[6] H. Y. Fan and J. R. Klauder, “Eigenvectors of two particles’ relative position and total mo-
mentum,” Phys. Rev. A 49, 704 (1994).
5
[7] A.Wu¨nsche,“Aboutintegrationwithinorderedproductsinquantumoptics,”J.Opt.B:Quan-
tum Semiclassical Opt. 1, R11 (1999).
[8] A. Einstein, B. Podolsky and N. Rosen, “Can Quantum-Mechanical Description of Physical
Reality Be Considered Complete?” Phys. Rev. 47, 777 (1935).
[9] H.-Y. Fan and H.-L. Lu, “2-mode Fresnel operator and entangled Fresnel transform”, Phys.
Lett. A 334, 132 (2005).
[10] D. F. V. Jamesand G. S. Agarwal,“The GeneralizedFresnelTransformand its Application to
Optics,” Opt. Commun. 126, 207 (1996).
[11] S. A. Collins, “Lens-system diffraction integral written in terms of matrix optic,” J. Opt. Soc.
Am. A 60, 1168 (1970).
[12] E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40,
749 (1932).
[13] J. Radon, “Uber die Bestimmung von Funktionen Durch Ihre Integralwerte Langs Gewisser
Mannigfaltigkeiten,” Ber. Verh. Saechs. Akad. Wiss. Leipzig Math. Phys. K1. 69, 262 (1917).
[14] Y. Zhang, B. Guo, B. Dong and G. Yang, “Optical implementations of the Radon– Wigner
display for one-dimensional signals,” Opt. Lett. 23, 1126 (1998).
[15] Wolfgang P. Schleich, Quantum Optics in Phase Space, (Wiley-VCH, Berlin, 2001) and refer-
ences therein.
[16] H.-Y.Fan,“TimeevolutionoftheWignerfunctionintheentangled-staterepresentation,”Phys.
Rev. A 65, 064102 (2002).
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