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Application of the two-mode squeezed coherent state representation in deriving generalized optical Collins formula Chuan-Mei Xie a,b,∗ , Hong-Yi Fan b a b

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

a r t i c l e

i n f o

a b s t r a c t We propose a new two-mode squeezed coherent state representation |z1 , z2 g which is characteristic of the correlation between the squeezing and the displacement. Based on it and using the technique of integration within an ordered product of operators we obtain a generalized two-mode Fresnel operator (GTFO), which is an image of the mapping from (z1 , z2 ) to (sz1 + rz2∗ , rz1∗ + sz2 ) in |z1 , z2 g representation. The matrix element of GTFO in the coordinate representation leads to a generalized two-dimensional Collins formula (Huygens–Fresnel integration transformation describing optical diffraction) in entangled form. © 2010 Elsevier GmbH. All rights reserved.

Article history: Received 23 December 2009 Accepted 14 June 2010

Keywords: Squeezed coherent state IWOP technique Generalized two-mode Fresnel operator Generalized two-dimensional Collins formula

1. Introduction In Fourier optical diffraction theory the two-dimensional Collins diffraction integral formula describing the light propagation in an optical system characterized by the [A, B; C, D] ray transfer matrix is [1], k exp (ikz) U2 (x2 , y2 ) = 2Bi

dx1 dy1 U1 (x1 , y1 ) × exp

ik 2B

A x12 + y12 − 2 (x1 x2 + y1 y2 ) + D x22 + y22

,

(1)

where AD − BC = 1. This formula reﬂects the Huygens–Fresnel principle of wave propagation [2]. In Ref. [3], based on the coherent state [4,5] |z representation and using the technique of integration within an ordered product (IWOP) [6,7] of operators, the single-mode Fresnel operator U1 (r, s) is derived as an image of the classical transformation from z → sz − rz∗ , U1 (r, s) ≡

√

s

d2 z sz − rz ∗ z| = exp

−

r 2s∗

a†2 exp

a† a +

1 2

ln

1 s∗

exp

r ∗ 2 2s∗

a

,

where a, a† = 1, (s, r), complex number with |s|2 − |r|2 = 1, are related to a classical ray transfer matrix

s=

(2)

1 [A + D − i (B − C)] , 2

r=−

1 [A − D + i (B + C)] , 2

A C

B D

by

(3)

The matrix element in the coordinate |x representation of U1 can lead to the transformation kernel of optical Collins formula connecting the input light ﬁeld f (x) and output light ﬁeld g (x ),

g x =

1 2piB

∞

exp −∞

i 2B

Ax2 − 2x x + Dx

2

f (x) dx,

∗ Corresponding author at: College of Physics & Material Science, Anhui University, Hefei 230039, China. Tel.: +86 13856059332. E-mail address: [email protected] (C.-M. Xie). 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.06.026

(4)

950

C.-M. Xie, H.-Y. Fan / Optik 122 (2011) 949–954

i.e.,

1 x U1 (r, s) |x =

exp

i 2B

2piB

Ax2 − 2x x + Dx

2

≡ k1M x , x .

(5)

M represents the matrix [A, B; C, D], Eq. (5) is the one-dimensional case of Eq. (1). If we use Dirac’s symbol to let f (x) = x| f , then Eq. (4) is expressed as

g x =

∞

x U1 (r, s) |x x| f dx = x U1 (r, s) f ,

(6)

−∞

which is just the quantum mechanical version of Fresnel transformation. The two-mode case was discussed in Ref. [4]. The obtained Fresnel operator and Collins diffraction integral formula have been widely applied to the relationship between classical optics and quantum optics [8–17]. In this paper we want to derive a new generalized two-dimensional (2D) Collins diffraction integral formula in the context of quantum optics, to be more concrete, we shall ﬁrstly construct the two-mode squeezing–displacement related squeezed-coherent state |z1 , z2 g representation, which is complete, g is an complex parameter, and then ﬁnd a generalized two-mode Fresnel operator (GTFO) by mapping from |z1 , z2 g → sz1 + rz2∗ , rz1∗ + sz2 . It is interesting to demonstrate that the matrix element of GTFO in the coordinate basis just lead to g

the generalized 2D Collins formula. Our work is arranged as follows: In Section 2, we propose the |z1 , z2 g representation, and prove its completeness relation by virtue of the IWOP technique. In Section 3, we derive the GTFO. In Section 4, we derive the new generalized 2D Collins diffraction integral formula in entangled form. In this theory, the relationship between quantum optics and classical optics is more clear. 2. The displacement–squeezing related squeezed two-mode coherent state Since the seventies of last century, increasing attention in the ﬁeld of quantum optics has been paid to squeezed states of light because of their potential uses in optical communication and interferometers. In this section we construct a new two-mode squeezed coherent state

2 2

z1, z2 = exp − |z1 | − |z2 | + a† fz1 + gz2∗ + b† gz1∗ + fz2 − 2fga† b† 00 g 2

(7)

2

expression of z1, z2 one can see that the displacement and squeezing are related. This state is complete, since g

where 00 is the vacuum state in Fock space, b, b† = 1, fg plays the role of squeezing parameter, with the constraint ff∗ + gg∗ = 1, in the

1 2

d2 z1 d2 z2 |z1 , z2 gg z1 , z2 | = 1,

(8)

† † 00 =: e−a a−b b : and the IWOP technique

which can be veriﬁed with the use of the normal ordering form 00

1 2 d z1 d2 z2 |z1 , z2 gg z1 , z2 | 2

=

|z2 |2 1 2 |z1 |2 − + a† fz1 + gz2∗ + b† gz1∗ + fz2 − 2fga† b† d z1 d2 z2 exp − 2 2 2

|z1 |2 |z2 |2 00 exp − − + a f ∗ z1∗ + g ∗ z2 + b g ∗ z1 + f ∗ z2∗ − 2f ∗ g ∗ ab

× 00

=

2 2 1 2 d z1 d2 z2 : exp{−|z1 |2 − |z2 |2 + z1 fa† + g ∗ b + z1∗ gb† + f ∗ a + z2 fb† + ag ∗ 2

∗

(9)

+z2 ga† + f ∗ b − 2fga† b† − 2f ∗ g ∗ ab − a† a − b† b} : = exp

fa† + g ∗ b

gb† + f ∗ a + fb† + ag ∗

ga† + f ∗ b − 2fga† b† − 2f ∗ g ∗ ab − a† a − b† b

= 1. Moreover, using the completeness relation of two-mode coherent state

2 2 2 2

˛, ˇ = exp − |˛| − |ˇ| + ˛a† + ˇb† 00 , d ˛d ˇ ˛, ˇ ˇ, ˛ = 1 2 2

2

and the overlap(11) ˛, ˇ z1 , z2 g = ˛ˇ exp −

+ ˛∗ a + ˇ∗ b × exp − + ˇ∗

gz1∗

+ fz2

|z1 |2 2

− 2fg˛∗ ˇ∗

−

|z2 |2 2

(10)

|z1 |2 2

−

|z2 |2 2

+ ˛∗ fz1 + gz2∗ + ˇ∗ gz1∗ + fz2 − 2fg˛∗ ˇ∗

we have

+ ˛∗ fz1 + gz2∗ + ˇ∗ gz1∗ + fz2 − 2fg˛∗ ˇ∗

00 = 00 exp − |˛|2 − |ˇ|2 2 2 |z1 |2 |z2 |2 ∗ ∗

00 = exp − |˛|2 − |ˇ|2 − 2 2

2

−

2

+˛

fz1 + gz2

C.-M. Xie, H.-Y. Fan / Optik 122 (2011) 949–954

z1 , z2 | z1 , z2 g =

g

z1 z2 |

d2 ˛d2 ˇ 2 2 ˛ˇ ˇ˛ z1 z2 g = e−|z1 | −|z2 | 2

d2 ˛d2 ˇ exp[−|˛|2 − |ˇ|2 + ˛ f ∗ z1∗ + g ∗ z2 + ˇ g ∗ z1 + f ∗ z2∗ 2

− 2f ∗ g ∗ ˛ˇ + ˛∗ fz1 + gz2∗ + ˇ∗ gz1∗ + fz2 − 2fg˛∗ ˇ∗ ] = where we have used the formula

d2 z exp{|z|2 + z + z ∗ + fz 2 + gz ∗2 } =

so the normalization of z1, z2

g

1

exp

2 − 4fg

951

1

2

(12)

1 − 4 fg

− + 2 g + 2 f 2 − 4fg

,

(13)

is

2 z1 , z2 | z1 , z2 g = 1. 1 − 4 fg

(14)

g

Due to

a|z1 , z2 g = fz1 + gz2∗ − 2fgb† |z1 , z2 g b|z1 z2 g = gz1∗ + fz2 − 2fga† |z1 , z2 g ,

(15)

|z1 , z2 g obeys the eigenvector equation

∗ 2 |z1 , z2 g = fz1 + gz2 |z1 , z2 g 1 − 4 fg ∗ b + 2fga† 2 |z1 , z2 g = gz1 + fz2 |z1 , z2 g . 1 − 4 fg a + 2fgb†

2

1 − 4 fg ≡ a , b + 2fga† /

If we let a + 2fgb† /

(16)

2

1 − 4 fg = b , then a , a † = 1, b , b

†

= 1, |z1 , z2 g is a two-mode squeezed state

indeed, but it differs from the usual one, since its squeezing parameter is related to its displacement parameter. 3. The generalized two-mode Fresnel operator in the |z1 , z2 g representation consider that the two-mode squeezed coherent state |z1 , z2 g has a movement in (z1 , z2 ) space characteristic of (z1 , z2 ) → We now ∗ ∗ sz1 + rz2 , rz1 + sz2 . Corresponding to it, we construct the following ket-bra integration operator,

U2g (r, s) = ss

1 2 z , z | , d z1 d2 z2 sz1 + rz2∗ , rz1∗ + sz2 gg 1 2 2

∗

(17)

where s and r are complex, satisfying |s|2 − |r|2 = 1,

(18)

which indicates that the movement is symplectic. Using Eq. (13) and the IWOP technique we can perform the integration in Eq. (17) and obtain

1 2 z , z | = ss∗ d z1 d2 z2 sz1 + rz2∗ , rz1∗ + sz2 2 gg 1 2

U2g (r, s) = ss∗

1 2 d z1 d2 z2 : exp{−|s|2 |z1 |2 − |s|2 |z2 |2 − sr ∗ z1 z2 − s∗ rz1∗ z2∗ 2

+ fsa† + gr ∗ a† + g ∗ b z1 + fsb† + gr ∗ b† + g ∗ a z2 + frb† + gs∗ b† + f ∗ a z1∗ + fra† + gs∗ a† + f ∗ b z2∗ − 2fga† b† − 2f ∗ g ∗ ab

− a† a − b† b}:= exp

f 2r g2r∗ + ∗ s s

Then with the help of the operator identity

exp l a† a + b† b we have

U2g (r, s) = exp

=: exp

el − 1

f 2r g2r∗ + s∗ s

a† a + b† b

† †

ab

a† b†

exp

†

: exp

|f |2

a† a + b† b

s∗

+

|g|2 −1 s

: × exp −

f ∗2 r ∗ g ∗2 r + ∗ s s

:

U2 (r, s) = s

ab .

(19)

(20)

†

a a + b b ln

|f |2 |g|2 + s∗ s

× exp −

f ∗2 r ∗ g ∗2 r + s∗ s

ab .

(21)

In particular, when g = 0 and f = 1 Eq. (21) reduces to √ U2g=0 (r, s) = s∗ U2 (r, s) where

1 2 d z1 d2 z2 sz1 + rz2∗ , rz1∗ + sz2 z1 , z2 | = exp 2

(22)

r s

a† b† ∗

exp

a† a + b† b + 1 ln

1 s∗

r∗

exp −

s

ab ∗

(23)

952

C.-M. Xie, H.-Y. Fan / Optik 122 (2011) 949–954

is the two-mode Fresnel operator in Ref. [4]; on the other hand, when g = 1, f = 0, √ U2g=1 (r, s) = sU2 (r ∗ , s∗ ) .

(24)

Thus we name Ug (r, s) the generalized two-mode Fresnel operator (GTFO). Further, using the IWOP technique and the formulas Eq. (2) and Eq. (1) we can verify †

−1 U2g (r, s) = U2g (r, s) ,

(25)

so U2g (r, s) is a unitary operator. Using the normally ordered form of U2g (r, s) in Eq. (19) we take its two-mode coherent state matrix element and immediately have

g2r∗ f 2r + s∗ s

z1 , z2 U2g (r, s) |z1 , z2 = exp{ −

|z1 |2 2

−

|z2 |2 2

z1∗ z2∗ −

g ∗2 r f ∗2 r ∗ + s∗ s

} = exp

∗ ∗ Y z1 z2

− Vz1 z2 + K

z1 z2 +

∗ z1 z1

|g|2 |f |2 + s∗ s

∗ + z2 z2

∗

∗

z1 z1 + z2 z2 −

|z1 |2 |z2 |2 − 2 2

|z |2 |z |2 |z1 |2 |z2 |2 − − − 1 − 2 2 2 2 2

(26)

where we have deﬁned K≡

|f |2 |g|2 + , ∗ s s

V≡

f ∗2 r ∗ g ∗2 r + , ∗ s s

Y≡

f 2r g2r∗ + , ∗ s s

(27)

Then, in the entangled state representation [18]

2

= exp − || + a† − ∗ b† + a† b† 00 ,

(28)

2

we can get the entangled state matrix element of U2g (r,s),

≡ U2g (r, s) =

=

d2 z1 d2 z2 d2 z1 d2 z2 4 d2 z1 d2 z2 d2 z1 d2 z2 4

| z1 , z2

z1 , z2 U2g (r, s) |z1 , z2 z1 , z2 |

2

2

exp −|z1 | − |z2 |

− |z1 |2

∗

− |z2 |2

∗

∗

||2 ||2 − − 2 2

∗

× exp −Vz1 z2 + z1∗ z2∗ + z1∗ + Kz1 z1 + Kz2 z2 + Y z1 z2 + z1 z2 + ∗ z1 − ∗ z2∗ − z2 =−

1

K K

− K −1

× exp

(1 + V ) (1 − Y )

−||2 K K + K −1 (1 + V ) (1 + Y ) − ||2 K K + K −1 (1 − Y ) (1 − V ) + 2∗ K + 2 ∗ K

−2K K − K −1 (1 + V ) (1 − Y )

(29)

By introducing

1 K + K −1 (1 − V ) (1 − Y ) , 2 i K − K −1 (1 + V ) (1 − Y ) , B= 2 i C=− K − K −1 (1 − V ) (1 + Y ) , 2 1 K + K −1 (1 + V ) (1 + Y ) , D= 2 A=

(30)

which just obeys AD − BC = 1,

(31)

then we can rewrite Eq. (29) as

≡ U2g (r, s) =

1 exp K2Bi

i 2B

A||2 + D||2 − (∗ + ∗ )

≡

(r,s) M , ≡ 2 , K 2 K

(32)

where the superscript M only means that the parameters of 2M are [A, B; C, D] and the subscript 2 means the two-dimensional kernel. Eq. (32) has the similar form as Eq. (5) except for its complex form. Taking 1 = x1 , 2 = x2 and 1 = x1 , 2 = x2 , we have

2M , = 2M x1 , x2 ; x1 , x2 = 1M x1 , x1 ⊗ 1M x2 , x2

(33)

This show that U2g (r, s) is really a generalized 2D Fresnel transformation operator. 4. The generalized 2D Collins diffraction integral formula in entangled form. Now we examine the matrix element of GTFO in coordinate eigenstates. Using Eq. (32) we have

x1 , x2 U2g (r, s) |x1 , x2 =

d2 d2 x1 , x2 | U2g (r, s) x1 , x2 , 2

(34)

C.-M. Xie, H.-Y. Fan / Optik 122 (2011) 949–954

953

where |x

1 , x2 is the two-mode coordinate eigenstate representation, using the Schmidt-decomposed relation of the bipartite entangled state (or ) [4],we can get √ √ −i 22 x1 , 1 + x2 − x1 · exp = 1 + i2 |x1 , x2 = exp (i1 2 ) ı 2 √ √ (35) x1 , x2 = 1 + i2 = exp (−i2 1 ) ı 21 + x2 − x1 · exp i 22 x1 . On substituting Eqs. (32) and (35) into Eq. (34) we derive

x1 , x2 U2g (r, s) |x1 , x2 =

1 4KiB

× exp

x +x i 2 d2 d2 x1 + x2 + exp i2 1√ 2 − i2 √ A2 + D22 − 22 2 2 2B 2 2

i A 2B

2

1 = exp √ 2K BC × exp

(x1 − x2 )2 +

i A

−i 4C

2B

2

D 2 x1 − x2 − (x1 − x2 ) x1 − x2 2

(x1 − x2 )2 +

A x1 + x2

2

D 2 x − x2 − (x1 − x2 ) x1 − x2 2 1

+ D(x1 + x2 )2 − 2 x1 + x2 (x1 + x2 )

.

(36)

In particular, if let g = 0 and f = 1 from Eqs. (27) and (30), we see A=

1 (s − r + s∗ − r ∗ ) , 2

B=

in this case Eq. (36) reduces to

C=−

i (s − r − s∗ + r ∗ ) , 2

D=

1 (s + r + s∗ + r ∗ ) , 2

x1 , x2 U2g (r, s) |x1 , x2 = s∗ x1 , x2 U2 (r, s) |x1 , x2 ,

where

i (s + r − s∗ − r ∗ ) , 2

1 exp √ 2 BC

x1 , x2 U2 (r, s) |x1 , x2 =

× exp

i A 2B

−i 4C

2

(x1 − x2 )2 +

A x1 + x2

2

(37)

(38)

2 D x1 − x2 − (x1 − x2 ) x1 − x2 2

+ D(x1 + x2 )2 − 2 x1 + x2 (x1 + x2 )

.

(39)

Eq. (39) is the entangled Fresnel transform in Ref. [4]. So, Eq. (36) is really a generalized two-dimensional Collins diffraction integral formula in entangled form. In summary, based on the displacement–squeezing related squeezed two-mode coherent state and using the technique of integration within an ordered product of operators we ﬁnd a generalized two-mode Fresnel operator (GTFO), whose matrix element in the coordinate representation leads to a generalized two-dimensional optical Collins formula. The GTFO corresponds to the variation of |z1 , z2 g from

|z1 , z2 g → sz1 + rz2∗ , rz1∗ + sz2 , this is a new example that there exists formal connection between classical optics and quantum optics. g

At this point we recall that in the history of quantum mechanics Schrödinger considered that classical dynamics of a point particle should be the “geometrical optics” approximation of a linear wave equation, in the same way as ray optics is a limiting approximation of wave optics. We expect that the generalized two-dimensional optical Collins formula may be realistically used in optical ray diffraction for propagation of light emitted from some complicated objects. Acknowledgements This work is supported by National Natural Science Foundation of China under grant 10874174 and the President Foundation of Chinese Academy of Science. References [1] S.A. Collins, Lens-system diffraction integral written in terms of matrix optics, J. Opt. Soc. Am. 60 (1970) 1168–1176. [2] J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, NewYork, 1972. [3] 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. [4] H.-y. Fan, H.-l. Lu, 2-Mode Fresnel operator and entangled Fresnel transform, Phys. Lett. A 334 (2005) 132–139. [5] J.R. Klauder, B.S. Skagerstam, Coherent States, WorldScientiﬁc, Singapore, 1985. [6] R.J. Glauber, Coherent and incoherent states of the radiation ﬁeld, Phys. Rev. 131 (1963) 2766–2788. [7] H.-y. Fan, H.-l. Lu, Y. Fan, Newton–Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations, Ann. Phys. 321 (2006) 480–494. [8] H.-y. Fan, Operator ordering in quantum optics theory and the development of Dirac’s symbolic method, J.Opt. B: Quan. Semiclass. Opt. 5 (2003) R147–R163. [9] 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. [10] S.-g. Liu, H.-y. Fan, Multiplication rule of similarity transformations in the coherent state representation and its mapping onto quantum optical generalized ABCD law, Opt. Int. J. Light Electron. Opt. (2009), doi:10.1016/j.ijleo.2009.02.028. [11] D.-m. Zhao, H.-d. Mao, D. Sun, Approximate analytical expression for the kurtosis parameter of off-axial Hermite-cosine-Gaussian beams propagating through apertured and misaligned ABCD optical systems, Optik 114 (2003) 535–538. [12] D.-m. Zhao, H.-d. Mao, D. Sun, S.-m. Wang, Approx-imate analytical representation of Wigner distribution function for Gaussian beams passing through ABCD optical systems with hard aperture, Optik 116 (2005) 211–218. [13] H.-y. Fan, L.-y. Hu, Fresnel-transform’s quantum correspondence and quantum optical ABCD law, Chin. Phys. Lett. 24 (2007) 2238–2241. [14] H.-y. Fan, L.-y. Hu, ABCD rule involved in the product of general SU(1,1) squeezing operators and quantum states Fresnel transformation, Opt. Commun. 281 (2008) 1629–1634.

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