Relation between quantum tomography and optical Fresnel transform 8 0 0 Hong-Yi Fan1,2and Li-yun Hu1∗ 2 1 Department of Physics, Shanghai Jiao Tong University,Shanghai200030, China n 2 a Department of Material Scienceand Engineering, Universityof Science and J Technology of China, Hefei, Anhui230026, China 3 1 February 2, 2008 ] h p - t Abstract n a Corresponding to optical Fresnel transformation characteristic of ray transfer matrix ele- u ments (A,B,C,D), AD−BC = 1, there exists Fresnel operator F(A,B,C,D) in quantum q optics,weshowthatundertheFresneltransformation thepurepositiondensity|xihx|becomes [ thetomographicdensity|xi hx|,whichisjusttheRadontransformoftheWigneroperator, s,rs,r 1 i.e. ∞ v F|xihx|F† =|xi hx|= dp′dx′δ x− Dx′−Bp′ ∆ x′,p′ , 0 s,rs,r Z Z −∞ ˆ ` ´˜ ` ´ 4 where s,r are the complex-value expression of (A,B,C,D). So the probability distribution for 9 theFresnelquadraturephaseisthetomography(RadontransformofWignerfunction),andthe 1 . tomogram of a state |ψi is just the wave function of its Fresnel transformed state F†|ψi, i.e. 1 s,rhx|ψi=hx|F†|ψi. Similarly, we find 0 8 ∞ 0 F|pihp|F† =|pis,rs,rhp|=Z Z dx′dp′δ p− Ap′−Cx′ ∆ x′,p′ . : −∞ ˆ ` ´˜ ` ´ v PACS numbers: 03.65.-w, 42.30.Kq i X r a 1 Introduction In quantum optics theory, all possible linear combination of quadratures X and P of the oscillator field mode a and a can be measured by the homodyne measurement just by varying the phase † of the local oscillator. The average of the random outcomes of the measurement, at a given local oscillatorphase,isconnectedwiththemarginaldistributionofWignerfunction,thusthehomodyne measurement of light field permits the reconstruction of the Wigner function of a quantum system by varying the phase shift between two oscillators. In Ref. [1] Vogel and Risken pointed out that the probability distribution for the rotated quadrature phase X [a exp(iθ)+aexp( iθ)]/√2, θ † ≡ − a,a = 1, which depends on only one θ angle, can be expressed in terms of Wigner function, † and that the reverse is also true (named as Vogel-Risken relation), i.e., one can obtain the Wigner (cid:2) (cid:3) distribution by tomographic inversionof a set of measuredprobability distributions, P (x ), of the θ θ quadratureamplitude. Smithey,BeckandRaymer[2,3,4]alsopointedoutthatoncethedistribution P (x )areobtained,onecanusetheinverseRadontransformationfamiliarintomographicimaging θ θ toobtainthe Wignerdistributionanddensitymatrix. The Radontransformofthe Wignerfunction is closely related to the expectation values or densities formed with the eigenstates to the rotated canonical observables. The field of problems of the reconstruction of the density operator from ∗Correspondingauthor. Email: [email protected] 1 such data is called quantum tomography. (Optical tomographic imaging techniques derive two- dimensional data from a three-dimensional object to obtain a slice image of the internal structure and thus have the ability to peer inside the object noninvasively.) The theoretical development in quantumtomographyinthelastdecadehasprogressedinthedirectionofdeterminingmorephysical relevant parameters of the density from tomographic data [5, 6, 7]. In[8,9]the RadontransformofWigner functionwhichdepends ontwocontinuousparametersis introduced. InthisLetterweextendtherotatedquadraturephaseX toX s a+ra +sa +r a /√2,where θ F ∗ † † ∗ ≡ A B s2 r2 =1, (s,r) are related to a classical ray transfer matrix (cid:0)by (cid:1) | | −| | C D (cid:18) (cid:19) 1 1 s= [A+D i(B C)], r = [A D+i(B+C)], AD BC =1. (1) 2 − − −2 − − We shall show that the (D,B) related Radon transform of the Wigner operator ∆(x,p) is just the pure state density operator x x formed with the eigenstates belonging to the quadrature | is,rs,rh | x . We name x the Fresnel transformed canonical observable, or Fresnel transformed quadrature F F phase, so the probability distribution for the Fresnel quadrature phase is the Radon transform of Wigner function. These can be expressed neatly by ∞ F|xihx|F† = |xis,rs,rhx|= dx′dp′δ[x−(Dx′−Bp′)]∆(x′,p′), Z Z−∞ 1 1 D = (s+s +r+r ), B = (s s+r r), (2) ∗ ∗ ∗ ∗ 2 2i − − where X x = x x , and x = F x , x is the eigenvector of X , and F is the Fresnel F | is,r | is,r | is,r | i | i θ=0 operator corresponding to classical Fresnel transform in optical diffraction theory which we shall describe in the following. We name x x the tomographicdensity operator. While the (A,C) | is,rs,rh | related Radon transform of ∆(x,p) is just the pure state density operator p p formed with | is,rs,rh | the eigenstates belonging to the conjugate quadrature of X , F ∞ F p p F = p p = dxdpδ[p (Dx Bp)]∆(x,p), | ih | † | is,rs,rh | ′ ′ − ′− ′ ′ ′ Z Z−∞ 1 1 A = (s r +s r), C = (s r s +r ). (3) ∗ ∗ ∗ ∗ 2 − − 2i − − Through(2)and(3)onecanseehowthequantumtomographyisrelatedtoopticalFresneltransform. TheopticaldiffractiontransformisdescribedbyFresnelintegrationwhoseparameters(A,B,C,D) are elements of a ray transfer matrix M describing optical systems, AD BC = 1, M belongs to − the unimodular symplectic group, the input light field f(x) and output light field g(x) are related ′ to each other by Fresnel integration [10, 11] g(x)= 1 ∞ exp i Ax2 2xx+Dx2 f(x)dx. (4) ′ ′ ′ √2πiB 2B − Z−∞ (cid:20) (cid:0) (cid:1)(cid:21) In order to find the quantum correspondence (Fresnel operator) of Fresnel transform, the coher- ent state [12, 13] x,p = exp[ 1 p2+x2 +(x+ip)a /√2] 0 is a good candidate for providing | i −4 † | i with classical phase-space description of quantum systems, thus we construct the following ket-bra (cid:0) (cid:1) projection operator 1 ∞ A B x x [A+D i(B C)] dxdp F, (5) 2 − − C D p p ≡ r ZZ (cid:12)(cid:18) (cid:19)(cid:18) (cid:19)(cid:29)(cid:28)(cid:18) (cid:19)(cid:12) (cid:12) (cid:12) −∞ (cid:12) (cid:12) (cid:12) (cid:12) as the FO, where the factor √.. is attached for anticipating the unitarity of the operator F. In A B x (5) the symplectic transformation mapping onto an Fresnel operator (FO) in C D p (cid:18) (cid:19)(cid:18) (cid:19) 2 Hilbert space is manifestly shown through the coherent state basis. Using the notation of z = | i exp 1 z 2+za 0 , z =(x+ip)/√2, and introducing complex numbers −2| | † | i (cid:2) (cid:3)1 1 s= [A+D i(B C)], r = [A D+i(B+C)], s2 r2 =1, (6) 2 − − −2 − | | −| | such that A B x s r z C D p = r∗ −s∗ z∗ ≡|sz−rz∗i (cid:12)(cid:18) (cid:19)(cid:18) (cid:19)(cid:29) (cid:12)(cid:18) − (cid:19)(cid:18) (cid:19)(cid:29) (cid:12) (cid:12) 1 (cid:12)(cid:12) = (cid:12)(cid:12)exp sz rz∗ 2+(sz rz∗)a† 0 , (7) −2| − | − | i (cid:20) (cid:21) so (5) becomes [14] d2z F (s,r)=√s sz rz z , (8) ∗ π | − ih | Z Using the vacuum-state projector 0 0 in normal ordering of boson operators | ih | 0 0 =: e−a†a: , (9) | ih | andthe technique ofintegrationwithin anorderedproduct(IWOP)ofoperators[15,16,17] wecan directly perform the integration in (8) and obtain d2z r s rs F(s,r) = √s : exp s2 z 2+sza†+z∗ a ra† + ∗ z2+ ∗z∗2 a†a : π −| | | | − 2 2 − Z (cid:20) (cid:21) 1 r 1 (cid:0) (cid:1) r = exp a 2 : exp 1 a a : exp ∗ a2 , (10) † † √s −2s s − 2s ∗ (cid:16) ∗ (cid:17) (cid:26)(cid:18) ∗ (cid:19) (cid:27) (cid:18) ∗ (cid:19) Note that this can be identified as a generalized squeezing operator with three real parameters [18, 19, 20]. It then follows 1 z 2 z 2 r r z z z F (s,r) z′ = exp | | | ′| z∗2 ∗ z′2+ ∗ ′ . (11) h | | i √s∗ "− 2 − 2 − 2s∗ − 2s∗ s∗ # Then using the overlap between the coordinate eigenvector and the coherent state x2 z2 z 2 x z =π−1/4exp +√2xz | | , (12) h | i − 2 − 2 − 2 ! andthecompletenessrelationofcoherentstateweobtainthematrixelementofF (s,r)incoordinate representation, d2z d2z ′ x F (s,r) x = x z z F (s,r) z z x ′ ′ 1 ′ ′ h | | i π h | ih | π | ih | i Z Z 1 i = exp Ax2 2xx+Dx2 , (13) ′ ′ √2πiB 2B − (cid:20) (cid:21) (cid:0) (cid:1) which is just the kernel of generalized Fresnel transform in (4). Now if we define g(x) = x g , f(x) = x f and using Eq. (13), we can rewrite Fresnel ′ ′ h | i h | i transform in Eq. (4) as ∞ x g = dx x F(A,B,C) x x f = x F (A,B,C) f . (14) ′ ′ ′ h | i h | | ih | i h | | i Z−∞ Therefore,the 1-dimensionalFT in classicaloptics correspondsto the 1-modeFO F (A,B,C) oper- ating on state vector f in Hilbert space, i.e. g =F (A,B,C) f . Using the Fock representation | i | i | i of x , | i 1 a 2 x =π 1/4exp x2+√2xa † 0 , (15) − † | i −2 − 2 | i (cid:26) (cid:27) 3 and (8) as well as (12) we calculate d2z F(s,r) x = √s sz rz∗ z x | i π | − ih | i Z d2z 1 = π 1/4√s exp sz rz 2+(sz rz )a 0 − ∗ ∗ † π −2| − | − | i Z (cid:20) (cid:21) x2 z 2 z 2 exp − 2 +√2xz∗− 2∗ − |2| !=|xis,r, (16) where π 1/4 s r x2 √2x s+r a 2 − ∗ ∗ † |xis,r ≡ √s +r exp{−s −+r 2 + s +r a†− s +r 2 }|0i. (17) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ We can see that x = F (s,r) x is the eigenstate of Fresnel transformed quadrature phase, | is,r | i because from √2x s+r a x = a x , (18) | is,r s∗+r∗ − s∗+r∗ †!| is,r so (s +r )a+(s+r)a x =√2x x . (19) ∗ ∗ † | is,r | is,r This can be further confirm(cid:2)ed by examining its com(cid:3)pleteness relation. Using the IWOP technique and(9)aswellas 1 = s∗ r∗ + s r ,wecanprovethat x makeupacompleteset(named s+r2 2(r∗−+s∗) 2(r−+s) | is,r | | as the tomography representation), ∞ ∞ dx s∗ r∗ s r x2 dx|xis,rs,rhx| = s+r √π: exp − s −+r + s+−r 2 −a†a Z−∞ Z−∞ | | (cid:26) (cid:18) ∗ ∗ (cid:19) a a s+r a 2 s +r a2 +√2x † + † ∗ ∗ : s +r s+r − s +r 2 − s+r 2 (cid:18) ∗ ∗ (cid:19) ∗ ∗ (cid:27) 2 ∞ dx 1 s∗a+ra†+sa†+r∗a = : exp x : =(210,) Z−∞ |s+r|√π (−|s+r|2 (cid:18) − √2 (cid:19) ) then using (6) and X = a+a†,P = ia† a as well as s +r = D+iB, s r = A iC, we can √2 √−2 ∗ ∗ ∗− ∗ − reform (20) as 1= 1 ∞ dx : exp 1 (x DX +BP)2 : , (21) √D2+B2 √π −D2+B2 − Z−∞ (cid:26) (cid:27) and x is express as | is,r π 1/4 A iC x2 √2x D iBa 2 − † |xis,r = √D+iB exp(−D−+iB 2 + D+iBa†− D+−iB 2 )|0i. (22) and (19) becomes (DX BP) x =x x . (23) − | is,r | is,r According to the Weyl quantization scheme [21] ∞ H(X,P)= dpdx∆(x,p)h(x,p), (24) Z Z−∞ where h(x,p) is the Weyl correspondence of H(X,P), ∆(x,p) is the Wigner operator [22, 23], ∆(x,p)= 1 ∞ dueipu x+ u x u . (25) 2π 2 − 2 Z−∞ (cid:12) ED (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 4 we know that the classical Weyl correspondence of the projection operator x x is | is,rs,rh | 2πTr ∆(x,p) x x = x ∞ dueip′u x + u x u x ′ ′ | is,rs,rh | s,rh | ′ 2 ′− 2 is,r h i Z−∞ (cid:12) ED (cid:12) 1 ∞ (cid:12) i (cid:12) = duexp i(cid:12)pu+ u(x Dx(cid:12) ) ′ ′ 2πB B − Z−∞ (cid:20) (cid:21) = δ[x (Dx′ Bp′)]. (26) − − From we see that under the Fresnel transformation (16) the pure position density x x becomes | ih | the tomographicdensity x x,andfurther from(24)and(26)wesee thatitis justthe Radon | is,rs,rh | transform of the Wigner operator, i.e. ∞ F x x F = x x = dxdpδ[x (Dx Bp)]∆(x,p). (27) | ih | † | is,rs,rh | ′ ′ − ′− ′ ′ ′ Z Z−∞ Therefore, the probability distribution for the Fresnel quadrature phase is the Radon transform of Wigner function x F ψ 2 = x ψ 2 = ∞ dxdpδ[x (Dx Bp)] ψ ∆(x,p) ψ . (28) † s,r ′ ′ ′ ′ ′ ′ |h | | i| | h | i| − − h | | i Z Z−∞ Moreover, the tomogram of quantum state ψ is just the wave function of its Fresnel transformed | i state F ψ , i.e. x ψ = x F ψ . (27) and (28) are the new relation between quantum † s,r † | i h | i h | | i tomography and optical Fresnel transform, which may provide experimentalists to figure out new approach for generating tomography. Similarly, we find that for momentum density, ∞ F p p F = p p = dxdpδ[p (Ap Cx)]∆(x,p), (29) | ih | † | is,rs,rh | ′ ′ − ′− ′ ′ ′ Z Z−∞ where π 1/4 D+iBp2 √2ip A+iC a 2 − † F p = p = exp + a + 0 . (30) | i | is,r √A iC (−A iC 2 A iC † A iC 2 )| i − − − − Asanapplicationoftherelation(27),werecallthattheFresneloperatorF(r,s)makesupaloyal representation of the symplectic group, i.e. d2z d2z F(r,s)F (r ,s)=√ss ′ sz rz z sz r z z ′ ′ ′ ′ π π | − ∗ih | ′ ′− ′ ′∗ih ′| Z Z 1 r 1 r = exp ′′ a 2 : exp 1 a a : exp ′′∗ a2 =F(r ,s ), (31) † † ′′ ′′ √s′′∗ (cid:18)−2s′′∗ (cid:19) (cid:26)(cid:18)s′′∗ − (cid:19) (cid:27) (cid:18)2s′′∗ (cid:19) where s′′ r′′ = s r s′ r′ , s 2 r 2 =1, (32) r′′∗ s′′∗ r∗ s∗ r′∗ s′∗ | ′′| −| ′′| (cid:18) − (cid:19) (cid:18) − (cid:19)(cid:18) − (cid:19) or A B A B A B AA +BC AB +BD ′′ ′′ = ′ ′ = ′ ′ ′ ′ . (33) C D C D C D AC+C D B C+DD ′′ ′′ ′ ′ ′ ′ ′ ′ (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) It then follows from (27) (33) and that F (r ,s)F(r,s) x x F (r,s)F (r ,s) ′ ′ ′ † ′† ′ ′ | ih | ∞ = dxdpδ[x ((B C+DD )x (AB +BD )p)]∆(x,p) ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ − − Z Z−∞ ∞ = dxdpδ[x (D x B p)]∆(x,p), (34) ′ ′ ′′ ′ ′′ ′ ′ ′ − − Z Z−∞ 5 in this way the complicated Radon transform of tomography can be viewed as the sequential oper- ation of two Fresnel transforms. In sum, we have revealed the new relation connecting optical Fresnel transformation to Radon transformationin quantum tomography,i.e. the probability distribution for the Fresnel quadrature phase is the tomography (Radon transform of Wigner function). The tomography representation x is set up, based on which the tomogramof a state ψ is just the wave function of its Fresnel s,r h | | i transformed state F ψ , i.e. x ψ = x F ψ . 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