The generalized Fresnel–Hadamard complementary transformation for asymmetric beamsplitter

Optik 123 (2012) 784–787

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The generalized Fresnel–Hadamard complementary transformation for asymmetric beamsplitter Chuan-mei Xie a,b,∗ , Hong-yi Fan b a b

College of Physics & Materials Science, Anhui University, Hefei 230039, China Department of Material Science and Engineering, University of Science and Technology of China, Hefei, Anhui 230026, China

a r t i c l e

i n f o

Article history: Received 19 December 2010 Accepted 30 April 2011

Keywords: Fresnel transformation Hadamard transformation IWOP technique Generalized Fresnel–Hadamard complementary transformation Asymmetric beamsplitter

a b s t r a c t Based on the newly developed parameterized coherent-entangled state representation we propose so-called the generalized Fresnel–Hadamard complementary transformation for asymmetric beamsplitter, which is unitary. The new unitary operator plays the role of both Fresnel transformation for a1 sin  − a2 cos  and Hadamard transformation for a1 cos  + a2 sin , respectively. Physically, a1 sin  − a2 cos  and a1 cos  + a2 sin  could be a asymmetric beamsplitter’s two output fields. We show that the two transformations are concisely expressed in the parameterized coherent-entangled state representation as a projective operator in integration form. © 2011 Elsevier GmbH. All rights reserved.

1. Introduction Optical transformation [1–8] is a major subject both in quantum optics and in classical optics, because each transformation may correspond to an optical instrument. For example, Fresnel transformation in classical optics can be realized by a lens system. In a preceding paper [9] we have found a unitary operator which causes a Fresnel–Hadamard complementary transformation in the context of quantum √ − a )/ 2 and simultaneously Hadamard transformation [9] optics, i.e., playing the role of both Fresnel transformation [5,8] for mode (a 2 1 √ √ √ for mode (a1 + a2 )/ 2, respectively , ai (i = 1, 2) is photon annihilator. Physically, (a1 − a2 )/ 2 and (a1 + a2 )/ 2 could be a symmetric beamsplitter’s two output fields when the two inputs are a1 and a2 , respectively. An interesting question thus naturally arises: if the beamsplitter is an asymmetric one, its scattering angle is , in this case the two output fields are a1 sin  − a2 cos  and a1 cos  + a2 sin , then what is the corresponding unitary operator responsible for a generalized Fresnel–Hadamard complementary transformation? In this paper, we shall construct a generalized Fresnel–Hadamard complementary transform which plays the role of both Fresnel transformation for mode a1 sin  − a2 cos  and Hadamard transformation for mode a1 cos  + a2 sin , respectively. These two modes could be two output fields of an asymmetric beamsplitter (for which the two input fields are a1 and a2 , respectively), and the asymmetric degree can be controllable. To find the generalized Fresnel–Hadamard complementary operator (GFHO) U, in Section 2, we shall introduce a kind of parameterized coherent-entangled state |z, x, and in Section 3, we assign U with a ket–bra projector in integration form. Then we use the technique of integration within an ordered product (IWOP) of operators to perform this integration to obtain U’s explicit form. Then we propose the so-called generalized Fresnel–Hadamard complementary transformation. As one can see later that the two transformations are concisely expressed in the |z, x, representation. In Section 4, we analyze the properties of the generalized Fresnel–Hadamard unitary operator and show that it plays the role of both Fresnel transformation for mode a1 sin  − a2 cos  and Hadamard transformation for mode a1 cos  + a2 sin , respectively.

∗ Corresponding author. E-mail address: [email protected] (C.-m. Xie). 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.04.025

C.-m. Xie, H.-y. Fan / Optik 123 (2012) 784–787

785

2. Brief review of the parameterized coherent-entangled state representation The parameterized coherent-entangled state representation is [10]



|z, x,

√ 1 x2 2 − |z|2 + ≡ exp − 2 4 



z x + 2



† a1

√ 2 + 



z x − 2





† a2

1 † † 2 − (a1 + a2 ) |00, 22

(1)

where x is a real number, z = z1 + iz2 is complex,  and  are two independent real parameters, =



2 + 2 ,

cos  =

 , 

sin  =

 

(2)

when  =  = 1, | z, x, reduces to |z, x, the coherent-entangled state representation used in Ref. [9]. From the basic commutation relations, Eq. (1) follows the common eigenvalue equations of (X1 + X2 ) and (a1 − a2 ), (X1 + X2 )|z, x, = x|z, x, z (a1 − a2 )|z, x, = √ |z, x, , 2

(3)

which also embodies the characteristic of both coherent state and entangled state, using the IWOP technique we can prove the completeness of |z, x, ,



∞

dx √ 

−∞



∞

−∞

d2 z |z, x,, z, x| = 1. 2

(4)

3. The generalized Fresnel–Hadamard transformation Based on the parameterized coherent-entangled representation |z, x, , we now construct the following ket–bra integration √ s U(, ; s, r) = √ 



d2 z 



∞



∞

dx dy exp −∞

−∞

 2ixy  2

|sz − rz ∗ , y,, z, x|,

(5)

we name U(, ;s, r) the generalized Fresnel–Hadamard combinatorial operator (GFHO). Substituting Eq. (1) into Eq. (5), we get U(, ; s, r)

=

√  2  ∞ ∞  2ixy  d z s dx dy exp √  −∞ −∞ 2   √  √    2 x z z 1 1 2 2 † † † † 2 ·exp − x + a1 + x − a2 − (a + a ) |0000| − |z|2 + 1 2 2 2   22  2 4  √  √    x2 z∗  z∗  1 1 2 2 2 ·exp − x + a1 + x − a2 − (a + a ) , − |z|2 + 1 2 2 4 2 2   22

(6)

Then using the IWOP technique and the two-mode vacuum projector’s normally ordered form |0000| =: exp[−a+ a − a+ a ] : and sepa1 1 2 2 rating the integration variances leads to U(, ; s, r)

=

  ∞ ∞ √  2ixy  y2 x2 s † dx dy exp exp − ˛ y + ˛x − + : √ 2 2 2  2 −∞ −∞  z∗ 1 r d z 2 2 ∗ 2 ∗ ∗2 † s exp − (2|s| |z| − sr z − rs z ) + ˇ z − z∗ + ˇ 4 2 2 2  1 †2 1 2 + + exp −a1 a1 − a2 a2 − ˛ − ˛ :, 4 4

(7)

where √

2 (a1 + a2 ) √ 2 ˇ= (a1 − a2 ),  ˛=

(8)

For the integration of d xdy, we can get



∞



∞

dx dy exp

−∞

=

 2ixy 



2

 exp

−

x2 y2 1 1 − + ˛† y + ˛x − ˛†2 − ˛2 2 2 4 4

 2 4 − 4 † † exp (a1 cos  + a2 sin )  4 + 4) 2( 4   + 4

+

−∞

2 2

4 − 4 2( 4 + 4)

2

(a1 cos  + a2 sin ) +



(9)



4i 2 † † (a cos  + a2 sin )(a1 cos  + a2 sin ) 4 + 4 1

,

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C.-m. Xie, H.-y. Fan / Optik 123 (2012) 784–787

While performing the integration over d2 z we need the formula



d2 z exp{ |z|2 + z + z ∗ + fz 2 + gz ∗2 }   1 −  + 2 g + 2 f =  , exp 2 − 4fg 2 − 4fg

(10)

and get









d2 z s 1 r z∗ exp − (2|s|2 |z|2 − sr ∗ z 2 − rs∗ z ∗2 ) + ˇ† z − z∗ + ˇ  4 2 2

 −r 

2 1 r∗ 2 1 † †2 =  exp ˇ + ˇ + ˇ ˇ 4s∗ 4s∗ 2s∗ (1/4)|s|2

  2 1 −r † † =  exp (a1 sin  − a2 cos ) 2s∗ (1/4)|s|2 +

(11)

1 † r∗ 2 † (a sin  − a2 cos )(a1 sin  − a2 cos ) + ∗ (a1 sin  − a2 cos ) s∗ 1 2s

.

So combined Eqs. (9) and (11), we can rewrite Eq. (5) as √ 1 s 2 2 U(; s, r) = √   2  4

−r+4 (1/4)|s| 2 1 † † † † : exp (a1 sin  − a2 cos ) + ∗ (a1 sin  − a2 cos )(a1 sin  − a2 cos ) 2s∗ s 1 1 r∗ 2 a − a+ a − ˛†2 − ˛2 + ∗ (a1 sin  − a2 cos ) − a+ 1 1 2 2 2s    4 4 2 4 − 4 4 − 4 4i 2 † 2 † † † (a + (a cos +a sin ) + (a cos +a sin ) + cos +a sin )(a cos +a sin ) : 1 2 1 2 1 2 2 2( 4 + 4) 2( 4 +4)  4 +4 1 Then after some rearrangement and calculations we finally get √ 1 s 2 2 U = √   2  4+4 (1/4)|s|      2 4 − 4 4i 2 † † † † : exp − 1 (a1 cos  + a2 sin )(a1 cos  + a2 sin ) (a1 cos  + a2 sin ) + 2( 4 + 4) 4 + 4









(12)

(13)

2 4 − 4 −r 2 † † + (a1 cos  + a2 sin ) + (a1 sin  − a2 cos ) 2s∗ 2( 4 + 4) 1 

r∗ 2 † † exp + ∗ (a1 sin  − a2 cos ) + − 1 (a sin  − a cos )(a sin  − a cos ) : 1 2 1 2 2s s∗

4. Properties For the commutation relationships †

†

[a1 cos  + a2 sin , a1 sin  − a2 cos ] = 0

(14)

and †

†

[a1 cos  + a2 sin , a1 cos  + a2 sin ] = 1, † † [a1 sin  − a2 cos , a1 sin  − a2 cos ] = 1

(15)

a1 sin  − a2 cos  can be considered a mode independent of another mode a1 cos  + a2 sin , thus we have the operator identity †

†

exp[f (a1 cos  + a2 sin )(a1 cos  + a2 sin )] 1 † † =: exp (e2f − 1)(a1 cos  + a2 sin )(a1 cos  + a2 sin ) : 2 † † exp[f (a1 sin  − a2 cos )(a1 sin  − a2 cos )] 1 † † (e2f − 1)(a1 sin  − a2 cos )(a1 sin  − a2 cos ) : =: exp 2

(16)

Using Eqs. (14) and (15) we can rewrite Eq. (13) as U = U1 U2

(17)

where U1 ()

=

1 √ exp s∗ 

·exp ·exp



 −r  2s∗

† (a1 ∗ r

2s∗

†

†

(a1 sin  − a2 cos ) †

2

sin  − a2 cos )(a1 sin  − a2 cos ) +

(a1 sin  − a2 cos )

2



1 1 ln ∗ 2 s

(18)

C.-m. Xie, H.-y. Fan / Optik 123 (2012) 784–787

and U2 ()



√ 4 



=

exp

 4+ 4

·exp

† (a1



·exp

cos 

4 − 4 2( 4 + 4)

† + a2

4 − 4 2( 4 + 4)

 †

†

(a1 cos  + a2 sin )



sin )(a1 cos  + a2 sin )ln





(a1 cos  + a2 sin )

2

787



4i 2 4 + 4

 (19)

2

while U1 () is the Fresnel operator for mode (a1 sin  − a2 cos ), U2 () is the Hadamard operator for mode (a1 cos  + a2 sin ). We see the generalized Fresnel–Hadamard complementary operator can play both the roles of Fresnel transformation (FT) for (a1 sin  − a2 cos ) and Hadamard transformation (HT) for (a1 cos  + a2 sin ). Physically, (a1 sin  − a2 cos ) and (a1 cos  + a2 sin ) can be two asymmetric output fields of an asymmetric beamsplitter [11] (the two input fields are a1 and a2 ) and the two output fields can be adjusted by the angle . In particular, when  = /4, the generalized Fresnel–Hadamard combinatorial operator (GFHO) √ reduces to the Fresnel–Hadamard √ complementary operator which for symmetric beamsplitter, i.e. the two output fields are (a1 − a2 )/ 2 and (a1 + a2 )/ 2; when  = /2, the two output fields are a1 and a2 . Eq. (17) then follows †

†

U(a1 sin  − a2 cos )U −1 = U1 (a1 sin  − a2 cos )U1−1 = s∗ (a1 sin  − a2 cos ) + r(a1 sin  − a2 cos ), † † † † † † U(a1 sin  − a2 cos )U −1 = U1 (a1 sin  − a2 cos )U1−1 = r ∗ (a1 sin  − a2 cos ) + s(a1 sin  − a2 cos ),

(20)

U(a1 cos  + a2 sin )U −1 = U1 (a1 cos  + a2 sin )U1−1 1 † † = [( 4 + 4)(a1 cos  + a2 sin ) − ( 4 − 4)(a1 cos  + a2 sin )], 4i 2 † † † † U(a1 cos  + a2 sin )U −1 = U1 (a1 cos  + a2 sin )U1−1 1 † † = [−( 4 + 4)(a1 cos  + a2 sin ) + ( 4 − 4)(a1 cos  + a2 sin )], 4i 2

(21)

√ from which we see the Hadamard–Fresnel complementary operator really plays the role of Fresnel transformation for (a1 − a2 )/ 2. Further, using Eq. (13) we have

which for the quadrature †

Xi =

ai + ai , √ 2

†

Pi =

ai − ai , √ 2i

(i = 1, 2),

(22)

leads to 2 (P1 cos  + P2 sin ), 2 2 = − 2 (X1 cos  + X2 sin ), 

U(X1 cos  + X2 sin )U −1 = U(P1 cos  + P2 sin )U −1

(23)

where (X1 cos  + X2 sin ) and (P1 cos  + P2 sin ) represent the center-of-mass position and center-of-mass momentum of the two particle system. From Eq. (23) we can see that the generalized Fresnel–Hadamard complementary operator can also play the role of exchanging the center-of-mass position – the center-of-mass momentum followed by a squeezing transform, with the squeezing parameter being  2 /2. It can be also seen that U is unitary, U+ U = UU+ = 1. In summary, by virtue of the IWOP technique we have introduced the generalized Fresnel–Hadamard complementary operator (GFHO). This unitary operator plays the role of both Fresnel transformation (FT) for mode (a1 sin  − a2 cos ) and Hadamard transformation (HT) for mode (a1 cos  + a2 sin ), respectively, and the two transformations are complementary. We have shown that the two transformations are concisely expressed in the parameterized coherent-entangled state representation as a projective operator in integration form. Physically, (a1 sin  − a2 cos ) and (a1 cos  + a2 sin ) can be two output fields of an asymmetric beamsplitter. If an optical device can be designed for the generalized Fresnel–Hadamard complementary transform, then it can be directly applied to these two output fields of the asymmetric beamsplitter. Acknowledgements This work is supported by the Doctoral scientific research Fund of Anhui University, China (Grant No. 33190059), the National Natural Science Foundation of China (Grant No. 10874174). References [1] J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, New York, 1972. [2] A. Belafhal, L. Dalil-Essakali, Collins formula and propagation of Bessel-modulated Gaussian light beams through an ABCD optical system, Opt. Commun. 177 (2000) 181–188. [3] H.-z. Liu, et al., Simple ABCD matrix method for evaluating optical coupling system laser diode to single-mode fiber with a lensed-tip, Optik 116 (2005) 415–418. [4] L. Mandl, E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, 1995. [5] H.-y. Fan, H.-l. Lu, Wave-function transformations by general SU(1,1) single-mode squeezing and analogy to Fresnel transformations in wave optics, Opt. Commun. 258 (2006) 51–58. [6] C.-m. Xie, H.-y. Fan, Application of the two-mode squeezed coherent state representation in deriving generalized optical Collins formula, Optik 122 (2010) 949–954. [7] H.-y. Fan, L.-y. Hu, Fresnel-transform’s quantum correspondence and quantum optical ABCD law, Chin. Phys. Lett. 24 (2007) 2238–2241. [8] H.-y. Fan, L.-y. Hu, Optical Fresnel transformation and quantum tomography, Opt. Commun. 282 (2009), 3734–373. [9] C.-m. Xie, H.-y. Fan, Fresnel–Hadamard complementary transformation in quantum optics, J. Mod. Opt. 57 (2010) 582–586. [10] L.-y. Hu, H.-y. Fan, A new bipartite coherent-entangled state generated by an asymmetric beamsplitter and its applications, J. Phys. B 40 (2007) 2099–2109. [11] M.A. Nielsen, I.L. Chuang, The Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000.