# Quantum Optical Version of Classical Optical Transformations and Beyond

Hong-yi Fan1 and Li-yun Hu2
1Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China;
Department of Material Science and Engineering, University of Science and
Technology of China, Hefei, Anhui 230026, China
2College of Physics & Communication Electronics, Jiangxi Normal University,
Nanchang 330022, China Corresponding author. E-mail address<EMAIL_ADDRESS>(L.Y. Hu)

###### Abstract

By virtue of the newly developed technique of integration within an ordered
product (IWOP) of operators, we explore quantum optical version of classical
optical transformations such as optical Fresnel transform, Hankel transform,
fractional Fourier transform, Wigner transform, wavelet transform and Fresnel-
Hadmard combinatorial transform etc. In this way one may gain benefit for
developing classical optics theory from the research in quantum optics, or
vice-versa. We can not only find some new quantum mechanical unitary operators
which correspond to the known optical transformations, deriving a new theorem
for calculating quantum tomogram of density operators, but also can reveal
some new classical optical transformations. For examples, we find the
generalized Fresnel operator (GFO) to correspond to the generalized Fresnel
transform (GFT) in classical optics. We derive GFO’s normal product form and
its canonical coherent state representation and find that GFO is the loyal
representation of symplectic group multiplication rule. We show that GFT is
just the transformation matrix element of GFO in the coordinate representation
such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian
beam propagation is directly demonstrated in the context of quantum optics.
Especially, the introduction of quantum mechanical entangled state
representations opens up a new area to finding new classical optical
transformations. The complex wavelet transform and the condition of mother
wavelet are studied in the context of quantum optics too. Throughout our
discussions, the coherent state, the entangled state representation of the
two-mode squeezing operators and the technique of integration within an
ordered product (IWOP) of operators are fully used. All these confirms Dirac’s
assertion: “ $...$for a quantum dynamic system that has a classical analogue,
unitary transformation in the quantum theory is the analogue of contact
transformation in the classical theory”.

Keywords: Dirac’s symbolic method; IWOP technique; entangled state of
continuum variables; entangled Fresnel transform; Collins formula; Generalized
Fresnel operator; complex wavelet transform; complex Wigner transform; complex
fractional Fourier transform; symplectic wavelet transform; entangled
symplectic wavelet transform; Symplectic-dilation mixed wavelet transform;
fractional Radon transform; new eigenmodes of fractional Fourier transform

###### Contents

  1. 1 Introduction
  2. 2 Some typical classical optical transformations
  3. 3 The IWOP technique and two mutually conjugate entangled states
    1. 3.1 The IWOP technique
    2. 3.2 The IWOP technique for deriving normally ordered Gaussian form of the completeness relations of fundamental quantum mechanical representations
    3. 3.3 Single-mode Wigner operator
    4. 3.4 Entangled state $\left|\eta\right\rangle$ and its Fourier transform in complex form
    5. 3.5 Two-mode Wigner operator in the $\left|\eta\right\rangle$ representation
  4. 4 Two deduced entangled state representations and Hankel transform
    1. 4.1 Deduced entangled states
    2. 4.2 Hankel transform between two deduced entangled state representations
    3. 4.3 Quantum optical version of classical circular harmonic correlation
  5. 5 Single-mode Fresnel operator as the image of the classical Optical Fresnel Transform
    1. 5.1 Single-mode GFO gained via coherent state method
    2. 5.2 Group Multiplication Rule for Single-mode GFO
  6. 6 Quantum Optical ABCD Law for optical propagation —single-mode case
  7. 7 Optical operator method studied via GFO’s decomposition
    1. 7.1 Four fundamental optical operators derived by decomposing GFO
    2. 7.2 Alternate decompositions of GFO
    3. 7.3 Some optical operator identities
  8. 8 Quantum tomography and probability distribution for the Fresnel quadrature phase
    1. 8.1 Relation between Fresnel transform and Radon transform of WF
    2. 8.2 Another new theorem to calculating the tomogram
  9. 9 Two-mode GFO and Its Application
    1. 9.1 Two-mode GFO gained via coherent state representation
    2. 9.2 Quantum Optical ABCD Law for two-mode GFO
    3. 9.3 Optical operators derived by decomposing GFO
      1. 9.3.1 GFO as quadratic combinations of canonical operators
      2. 9.3.2 Alternate decompositions of GFO and new optical operator identities
    4. 9.4 Quantum tomography and probability distribution for the Fresnel quadrature phase—two-mode entangled case
      1. 9.4.1 $\left|\eta\right\rangle_{s,r\text{ }s,r}\left\langle\eta\right|$ as Radon transform of the entangled Wigner operator
      2. 9.4.2 Inverse Radon transformation
  10. 10 Fractional Fourier Transformation (FrFT) for 1-D case
    1. 10.1 Quantum version of FrFT
    2. 10.2 On the Scaled FrFT Operator
    3. 10.3 An integration transformation from Chirplet to FrFT kernel
  11. 11 Complex Fractional Fourier Transformation
    1. 11.1 Quantum version of CFrFT
    2. 11.2 Additivity property and eigenmodes of CFrFT
    3. 11.3 From Chirplet to CFrFT kernel
      1. 11.3.1 New complex integration transformation
      2. 11.3.2 Complex integration transformation and complex Weyl transformation
    4. 11.4 Squeezing for the generalized scaled FrFT
      1. 11.4.1 Quantum correspondence of the scaled FrFT
      2. 11.4.2 The Scaled CFrFT
      3. 11.4.3 Entangled scaled FrFT
  12. 12 Adaption of Collins diffraction formula and CFrFT
    1. 12.1 Adaption of the Collins formula to CFrFT
    2. 12.2 Adaption of the additivity property of CFrFT to the Collins formula for two successive Fresnel diffractions
  13. 13 The Fractional Radon transform
  14. 14 Wavelet transformation and the IWOP technique
    1. 14.1 Quantum optical version of classical WTs
    2. 14.2 The condition of mother wavelet in the context of quantum optics
    3. 14.3 Quantum mechanical version of Parseval theorem for WT
    4. 14.4 Inversion formula of WT
    5. 14.5 New orthogonal property of mother wavelet in parameter space
    6. 14.6 WT and Wigner-Husimi Distribution Function
  15. 15 Complex Wavelet transformation in entangled state representations
    1. 15.1 CWT and the condition of Mother Wavelet
    2. 15.2 Parseval Theorem in CWT
    3. 15.3 Orthogonal property of mother wavelet in parameter space
    4. 15.4 CWT and Entangled Husimi distribution
      1. 15.4.1 CWT and its quantum mechanical version
      2. 15.4.2 Relation between CWT and EHDF
  16. 16 Symplectic Wavelet transformation (SWT)
    1. 16.1 Single-mode SWT
      1. 16.1.1 Properties of symplectic-transformed—translated WT
      2. 16.1.2 Relation between $W_{\psi}f\left(r,s;\kappa\right)$ and optical Fresnel transform
    2. 16.2 Entangled SWT
    3. 16.3 Symplectic-dilation mixed WT
  17. 17 Fresnel-Hadamard combinatorial transformation
    1. 17.1 The Hadamard-Fresnel combinatorial operator
    2. 17.2 The properties of Hadamard-Fresnel operator

##  1 Introduction

The history of quantum mechanics records that from the very beginning the
founders of the quantum theory realized that there might exist formal
connection between classical optics and quantum mechanics. For example,
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; Schrödinger also
searched for some quantum mechanical state which behaves like a classical
‘particle’, and this state was later recognized as the coherent state [1, 2,
3], which plays an essential role in quantum optics theory and laser physics;
As Dirac wrote in his famous book $<$Principles of Quantum Mechanics$>$[4]:
“$\cdot\cdot\cdot$ for a quantum dynamic system that has a classical analogue,
unitary transformation in the quantum theory is the analogue of contact
transformation in the classical theory”. According to Dirac, there should
exist a formal correspondence between quantum optics unitary-transform
operators and classical optics transformations. Indeed, in the last century
physicists also found some rigorous mathematical analogies between classical
optics and quantum mechanics, i.e. the similarity between the optical
Helmholtz equation and the time-independent Schrödinger equation; Since 1960s,
the advent of a laser and the appearance of coherent state theory of radiation
field [1, 2, 5], quantum optics has experienced rapid development and achieved
great success in revealing and explaining the quantum mechanical features of
optical field and non-classical behavior (for instance, Hanbury-Brown-Twiss
effect, photon antibunching, squeezing, sub-Poissonian photon statistics) of
photons in various photon-atom interactions [6]. The relationship between
classical and quantum coherence has been discussed in the book of Mandel and
Wolf [6]; The Hermite-Gauss or the Laguerre-Gauss modes of a laser beam are
described using the bosonic operator algebra by Nienhuis and Allen [7]. In
addition, displaced light beams refracted by lenses according to the law of
geometrical optics, were found to be the paraxial optics analog of a coherent
state. Besides, phase space correspondence between classical optics and
quantum mechanics, say for example, the Wigner function theory, is inspected
in the literature [8].

On the other hand, classical optics, which tackles vast majority of physical-
optics experiments and is based on Maxwell’s equations, has never ceased its
own evolving steps, physicists have endeavored to develop various optical
transforms in light propagation through lens systems and various continuous
media. The two research fields, quantum optics and classical optics, have
their own physical objects and conceptions. From the point of view of
mathematics, classical optics is framed in the group transform and associated
representations on appropriate function space, while quantum optics deals with
operators and state vectors, and their overlap seems little at first glance.
It seems to us that if one wants to further relate them to each other, one
needs some new theoretical method to ”bridge” them. For example, what is the
quantum mechanical unitary operator corresponding to the Fresnel transform in
Fourier optics? Is there any so-called Fresnel operator as the image of
classical generalized Fresnel transform? Since generalized Fresnel transforms
are very popularly used in optical instrument design and optical propagation
through lenses and various media, it is worth of studying these transforms in
the context of quantum optics theory, especially based on coherent state,
squeezed state [9, 10] and the newly invented entangled state theory [11, 12,
13, 14].

Fortunately, the recently developed technique of integration within an ordered
product (IWOP) of operators [15, 16, 17] is of great aid to studying quantum
optical version of classical optical transformations. Using the IWOP technique
one may gain benefit for classical optics from quantum optics’ research, or
vice-versa.

Our present Review is arranged as follows: in section 2 we briefly recall the
classical diffraction theory [18, 19], this is preparing for later sections in
which we shall show that most frequently employed classical optical transforms
have their counterparts in quantum optics theory. In section 3 we introduce
the IWOP technique and demonstrate that the completeness relation of
fundamental quantum mechanical representations can be recast into normally
ordered Gaussian operator form. Using the IWOP technique we can directly
perform the asymmetric ket-bra integration
$\mu^{-1/2}\int_{-\infty}^{\infty}dq\left|q/\mu\right\rangle\left\langle
q\right|$ in the coordinate representation, which leads to the normally
ordered single-mode squeezing operator, this seems to be a direct way to
understanding the squeezing mechanism as a mapping from the classical scaling
$q\rightarrow q/\mu$. In section 4 with the help of IWOP technique and based
on the concept of quantum entanglement of Einstein-Podolsky-Rosen [20] we
construct two mutually conjugate entangled states of continuum variables,
$\left|\eta\right\rangle$ versus $\left|\xi\right\rangle,$ and their deduced
entangled states (or named correlated-amplitude—number-difference entangled
states), they are all qualified to make up quantum mechanical representations.
It is remarkable that using the IWOP technique to performing the asymmetric
ket-bra integration $\mu^{-1}\int
d^{2}\eta\left|\eta/\mu\right\rangle\left\langle\eta\right|$ leads to the two-
mode normally ordered two-mode squeezing operator, this implies that the two-
mode squeezed state is simultaneously an entangled state. We point out that
the entangled state $\left|\eta\right\rangle$ also embodies entanglement in
the aspect of correlative amplitude and the phase. We are also encouraged that
the overlap between two mutually conjugate deduced entangled states is just
the Bessel function— the optical Hankel transform kernel [21], which again
shows that the new representations in the context of physics theory match
beautiful mathematical formalism exactly. We then employ the deduced entangled
states to derive quantum optical version of classical circular harmonic
correlation. Section 5 is devoted to finding a quantum operator which
corresponds to the optical Fresnel transform, with use of the coherent state
representation and by projecting the classical sympletic transform
$z\rightarrow sz-rz^{\ast}$ ($\left|s\right|^{2}-\left|r\right|^{2}=1)$ in
phase space onto the quantum mechanical Hilbert space, we are able to
recognize which operator is the single-mode Fresnel operator (FO). It turns
out that the 1-dimensional optical Fresnel transform is just the matrix
element of the Fresnel operator $F$ in the coordinate eigenstates. Besides,
the coherent state projection operator representation of FO constitutes a
loyal realization of symplectic group, which coincides with the fact that two
successive optical Fresnel transforms make up a new Fresnel transform. Then in
Section 6 based on the coherent state projection representation of FO, we
prove $ABCD$ rule for optical propagation in the context of quantum optics. In
section 7 the quadratic operator form of FO is also presented and the four
fundamental optical operators are derived by decomposing the FO. In section 8
we discuss how to apply the Fresnel operator to quantum tomography theory, by
introducing the Fresnel quadrature phase $FXF^{\dagger}=X_{F},$ we point out
that Wigner operator’s Radon transformation is just the pure state projection
operator $\left|x\right\rangle_{s,rs,r}$ $\left\langle x\right|$, where
$\left|x\right\rangle_{s,r}=F\left|x\right\rangle$ and $\left|x\right\rangle$
is the position eigenstate, so the probability distribution for the Fresnel
quadrature phase is the Radon transform of the Wigner function. Moreover, the
tomogram of quantum state $\left|\psi\right\rangle$ is just the squared
modulus of the wave function ${}_{s,r}\left\langle
x\right|\left.\psi\right\rangle.$ This new relation between quantum tomography
and optical Fresnel transform may provide experimentalists to figure out new
approach for testing tomography. In addition, we propose another new theorem
for calculating tomogram, i.e., the tomogram of a density operator $\rho$ is
equal to the marginal integration of the classical Weyl correspondence
function of $F^{\dagger}\rho F$. In section 9 by virtue of the coherent state
and IWOP method we propose two-mode generalized Fresnel operator (GFO), in
this case we employ the entangled state representation to relate the 2-mode
GFO to classical transforms, since the 2-mode GFO is not simply the direct
product of two 1-mode GFOs. The corresponding quantum optics $ABCD$ rule for
two-mode case is also proved. The 2-mode GFO can also be expressed in
quadratic operators form in entangled way. The relation between optical FT and
quantum tomography in two-mode case is also revealed. In section 10 we propose
a kind of integration transformation,
$\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}h(p,q)\equiv
f\left(x,y\right),$ which is invertible and obeys Parseval theorem.
Remarkably, it can convert chirplet function to the kernel of fractional
Fourier transform (FrFT). This transformation can also serve for solving some
operator ordering problems. In section 11 we employ the entangled state
representation to introduce the complex FrFT (CFrFT), which is not the direct
product of two independent 1-dimensional FrFT transform. The eigenmodes on
CFrFT is derived. New eigen-modes for light propagation in graded-index medium
and the fractional Hankel transform are presented. The Wigner transform theory
is extended to the complex form and its relation to CFrFT is shown; The
integration transformation in section 10 is also extended to the entangled
case. In section 12 we shall treat the adaption problem of Collins diffraction
formula to the CFrFT with the use of two-mode (3 parameters) squeezing
operator and in the entangled state representation of continuous variables, in
so doing the quantum mechanical version of associated theory of classical
diffraction and classical CFrFT is obtained, which connects classical optics
and quantum optics in this aspect. In section 13 we introduce a convenient way
for constructing the fractional Radon transform. the complex fractional Randon
transform is also proposed; In sections 14 and 15 we discuss quantum optical
version of classical wavelet transforms (WTs), including how to recast the
condition of mother wavelet into the context of quantum optics; how to
introduce complex wavelet transform with use of the entangled state
representations. Some properties, such as Parseval theorem, Inversion formula,
and orthogonal property, the relation between WT and Wigner-Husimi
distribution function are also discussed. In section 16, we generalize the
usual wavelet transform to symplectic wavelet transformation (SWT) by using
the coherent state representation and making transformation $z\rightarrow
s\left(z-\kappa\right)-r\left(z^{\ast}-\kappa^{\ast}\right)$
($\left|s\right|^{2}-\left|r\right|^{2}=1)$ in phase space. The relation
between SWT and optical Fresnel transformation is revealed. Then the SWT is
extended to the entangled case by mapping the classical mixed transformation
$\left(z,z^{\prime}\right)\rightarrow\left(sz+rz^{\prime\ast},sz^{\prime}+rz^{\ast}\right)$
in 2-mode coherent state $\left|z,z^{\prime}\right\rangle$ representation. At
the end of this section, we introduce a new symplectic-dilation mixed WT by
employing a new entangled-coherent state representation
$\left|\alpha,x\right\rangle$. The corresponding classical optical transform
is also presented. In the last section, we introduce the Fresnel-Hadamard
combinatorial operator by virtue of the IWOP technique and
$\left|\alpha,x\right\rangle$. This unitary operator plays the role of both
Fresnel transformation for mode $\frac{a_{1}-a_{2}}{\sqrt{2}}$ and Hadamard
transformation for mode $\frac{a_{1}+a_{2}}{\sqrt{2}},$ respectively, and the
two transformations are combinatorial. All these sections are used to prove
the existence of a one-to-one correspondence between quantum optical operators
that transform state vectors in Hilbert space and the classical optical
transforms that change the distribution of optical field.

##  2 Some typical classical optical transformations

Here we briefly review some typical optical transforms based on light
diffraction theory. These transformations, as one can see in later sections,
are just the correspondence of some representation transformations between
certain quantum mechanical states of which some are newly constructed.

It was Huygens who gave a first illustrative explanation to wave theory by
proposing every point in the propagating space as a sub-excitation source of a
new sub-wave. An intuitive theory mathematically supporting Huygens’ principle
is the scalar diffraction approximation, so named because optical fields
(electromagnetic fields) actually are vector fields, whereby the theory is
valid approximately. This theory is based on the superposition of the combined
radiation field of multiple re-emission sources initiated by Huygens. Light
diffraction phenomena has played an important role in the development of the
wave theory of light, and now underlies the Fourier optics and information
optics. The formulation of a diffraction problem essentially considers an
incident free-space wave whose propagation is interrupted by an obstacle or
mask which changes the phase and/or amplitude of the wave locally by a well
determined factor [22]. A more rigorous, but still in the scheme of scalar
wave, derivation has been given by Kirchhoff who reformulated the diffraction
problem as a boundary-value problem, which essentially justifies the use of
Huygens principle. The Fresnel-Kirchhoff (or Rayleigh-Sommerfeld) diffraction
formula is practically reduced to the Fresnel integral formula in paraxial and
far-field approximation [18, 19] that reads:

$U_{2}\left(x_{2},y_{2}\right)=\frac{\exp\left(ikz\right)}{i\lambda
z}\int\int_{-\infty}^{\infty}U_{1}\left(x_{1},y_{1}\right)\exp\left\\{i\frac{k}{2z}\left[\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}\right]\right\\}dx_{1}dy_{1},$
(1)

where $U_{1}\left(x_{1},y_{1}\right)$ is the optical distribution of a
2-dimensional light source and $U_{2}\left(x_{2},y_{2}\right)$ is its image on
the observation plane, $\lambda$ is the optical wavelength,
$k=\frac{2\pi}{\lambda}$ is the wave number in the vacuum and $z$ is the
propagation distance. When

$z^{2}\gg\frac{k}{2}\left(x_{1}^{2}+y_{1}^{2}\right)_{\max},$ (2)

is satisfied, Eq. (1) reduces to

$\displaystyle U_{2}\left(x_{2},y_{2}\right)$
$\displaystyle=\frac{\exp\left(ikz\right)\exp\left[i\frac{k}{2z}\left(x_{1}^{2}+y_{1}^{2}\right)\right]}{i\lambda
z}$
$\displaystyle\times\int\int_{-\infty}^{\infty}U_{1}\left(x_{1},y_{1}\right)\exp\left[-i\frac{2\pi}{\lambda
z}\left(x_{1}x_{2}+y_{1}y_{2}\right)\right]dx_{1}dy_{1},$ (3)

which is named the Fraunhofer diffraction formula.

The Fresnel integral is closely related to the fractional Fourier transform
(FrFT), actually, it has been proved that the Fresnel transform can be
interpreted as a scaled FrFT with a residual phase curvature [23]. The FrFT is
a very useful tool in Fourier optics and information optics. This concept was
firstly introduced in 1980 by Namias [24] but not brought enough attention
until FrFT was defined physically, based on propagation in quadratic graded-
index media (GRIN media). Mendlovic and Ozaktas [25, 26] defined the
$\alpha$th FrFT as follows: Let the original function be input from one side
of quadratic GRIN medium, at $z=0$. Then, the light distribution observed at
the plane $z=z_{0}$ corresponds to the $\alpha$ equal to the ($z_{0}/L$)th
fractional Fourier transform of the input fraction, where
$L\equiv(\pi/2)(n_{1}/n_{2})^{1/2}$ is a characteristic distance. The FrFT can
also be implemented by lenses. Another approach for introducing FrFT was made
by Lohmann who pointed out the algorithmic isomorphism among image rotation,
rotation of the Wigner distribution function [29], and fractional Fourier
transforming [27]. Lohmann proposed the FrFT as the transform performed on a
function that leads to a rotation with an angle of the associated Wigner
distribution function, in this sense, the FrFT bridges the gap between
classical optics and optical Wigner distribution theory. Recently, the FrFT
has been paid more and more attention within different contexts of both
mathematics and physics [24, 25, 26, 27, 28]. The FrFT is defined as

$\displaystyle\mathcal{F}_{\alpha}\left[U_{1}\right]\left(x_{2},y_{2}\right)=\frac{e^{i(1-\alpha)\frac{\pi}{2}}}{2\sin\left(\frac{\pi}{2}\alpha\right)}\exp\left[-\frac{i\left(x_{2}^{2}+y_{2}^{2}\right)}{2\tan\left(\frac{\pi}{2}\alpha\right)}\right]$
$\displaystyle\times\int\int_{-\infty}^{\infty}\frac{dx_{1}dy_{1}}{\pi}\exp\left[-\frac{i\left(x_{1}^{2}+y_{1}^{2}\right)}{2\tan\left(\frac{\pi}{2}\alpha\right)}\right]\exp\left[\frac{i\left(x_{2}x_{1}+y_{2}y_{1}\right)}{\sin\left(\frac{\pi}{2}\alpha\right)}\right]U_{1}\left(x_{1},y_{1}\right).$
(4)

We can see that $F_{0}$ is the identity operator and $F_{\pi/2}$ is just the
Fourier transform. The most important property of FRFT is that $F_{\alpha}$
obeys the semigroup property, i.e. two successive FrFTs of order $\alpha$ and
$\beta$ makes up the FrFT of order $\alpha+\beta.$ A more general form
describing the light propagation in an optical system characterized by the
$\left[A,B;C,D\right]$ ray transfer matrix is the Collins diffraction integral
formula [30],

$\displaystyle U_{2}\left(x_{2},y_{2}\right)$
$\displaystyle=\frac{k\exp\left(ikz\right)}{2\pi
Bi}\int\int_{-\infty}^{\infty}dx_{1}dy_{1}U_{1}\left(x_{1},y_{1}\right)$
$\displaystyle\times\exp\left\\{\frac{ik}{2B}\left[A\left(x_{1}^{2}+y_{1}^{2}\right)-2\left(x_{1}x_{2}+y_{1}y_{2}\right)+D\left(x_{2}^{2}+y_{2}^{2}\right)\right]\right\\}dx_{1}dy_{1},$
(5)

where $AD-BC=1$ if the system is lossless. One can easily find the similarity
between Collins formula and the FrFT by some scaling transform and relating
the $\left[A,B,C,D\right]$ matrix to $\alpha$ in the FrFT [31]. Note that
$M=\left(\begin{array}[c]{cc}A&B\\\ C&D\end{array}\right)$ is a ray transfer
matrix describing optical systems belonging to the unimodular symplectic
group. When treating the light propagation in optical elements in near-axis
approximation, matrices $M$ representing linear transformations are a
convenient mathematical tool for calculating the fundamental properties of
optical systems, which is the origin of the name of matrix optics. In
cylindrical coordinates the Collins formula is expressed as [30, 32]

$U_{2}\left(r_{2},\varphi\right)=\frac{i}{\lambda
B}{\int}_{0}^{\infty}{\int}_{0}^{2\pi}\exp\left\\{-\frac{i\pi}{\lambda
B}\left[Ar_{1}^{2}+Dr_{2}^{2}-2r_{1}r_{2}\cos\left(\theta-\varphi\right)\right]\right\\}U_{1}\left(r_{1},\theta\right)r_{1}dr_{1}d\theta$
(6)

where $x_{1}=r_{1}\cos\theta,$ $y_{1}=r_{1}\sin\theta,$
$x_{2}=r_{2}\cos\varphi$ and $y_{2}=r_{2}\sin\varphi.$ When
$U_{1}\left(r_{1},\theta\right)$ has rotational symmetry

$U_{1}\left(r_{1},\theta\right)=u_{1}\left(r_{1}\right)\exp\left(im\theta\right),\text{
}U_{2}\left(r_{2},\varphi\right)=u_{2}\left(r_{2}\right)\exp\left(im\varphi\right),$
(7)

then (6) becomes

$u_{2}\left(r_{2}\right)=\frac{2\pi}{\lambda
B}\exp\left[i\left(1+m\right)\frac{\pi}{2}\right]{\int}_{0}^{\infty}\exp\left[-\frac{i\pi}{\lambda
B}\left[Ar_{1}^{2}+Dr_{2}^{2}\right]\right]J_{m}\left(\frac{2\pi
r_{1}r_{2}}{\lambda B}\right)u_{1}\left(r_{1}\right)r_{1}dr_{1},$ (8)

where we have used the $m$-order Bessel function

$J_{m}\left(x\right)=\frac{1}{2\pi}{\int}_{0}^{2\pi}\exp\left[ix\cos\theta+im\left(\theta-\frac{\pi}{2}\right)\right]d\theta.$
(9)

When $A=0$, (8) reduces to the standard Hankel transform (up to a phase
factor)

$u_{2}\left(r_{2}\right)\rightarrow\frac{2\pi}{\lambda
B}{\int}_{0}^{\infty}J_{m}\left(\frac{2\pi r_{1}r_{2}}{\lambda
B}\right)u_{1}\left(r_{1}\right)r_{1}dr_{1},$ (10)

The compact form of one-dimensional Collins formula is

$g\left(x_{2}\right)=\int_{-\infty}^{\infty}\mathcal{K}^{M}\left(x_{2},x_{1}\right)f\left(x_{1}\right)dx_{1},$
(11)

where the transform kernel is

$\mathcal{K}^{M}\left(x_{2},x_{1}\right)=\frac{1}{\sqrt{2\pi
iB}}\exp\left[\frac{i}{2B}\left(Ax_{1}^{2}-2x_{2}x_{1}+Dx_{2}^{2}\right)\right],$
(12)

$M$ is the parameter matrix $\left[A,B,C,D\right]$. Eq. (12) is called
generalized Fresnel transform [33, 34, 35, 36]. In the following sections we
will show how we find the quantum optical counterpart for those
transformations of classical optics. For this purpose in the next chapter we
introduce the IWOP technique to demonstrate how Dirac’s symbolic method can be
developed and be applied to quantum optics theory. Also, we briefly review
some properties of the entangled state [11, 12, 13, 14] and reveal the
connection between the mutual transform generated by these entangled states
and the Hankel transform in classical optics.

##  3 The IWOP technique and two mutually conjugate entangled states

###  3.1 The IWOP technique

The history of mathematics tells us that whenever there appears a new
important mathematical symbol, there coexists certain operational rules for
it, the quantum mechanical operators in ket-bra projective form (the core of
Dirac’s symbolic method) also need their own operational rules. The
terminology “symbolic method” was first shown in the preface of Dirac’s book
$<$The Principle of Quantum Mechanics$>$: “The symbolic method, which deals
directly in an abstract way with the quantities of fundamental
importance$\cdot\cdot\cdot$, however, seems to go more deeply into the nature
of things. It enables one to express the physical law in a neat and concise
way, and will probably be increasingly used in the future as it becomes better
understood and its own special mathematics gets developed.” [4] Then two
questions naturally arise: How to better understand the symbolic method? How
to develop Dirac’s symbolic method, especially its mathematics? We noticed
that Newton-Leibniz integration rule only applies to commuting functions of
continuum variables, while operators made of Dirac’s symbols (ket versus bra,
e.g., $\left|q/\mu\right\rangle\left\langle q\right|$ of continuous parameter
$q$) in quantum mechanics are usually not commutative. Therefore integrations
over the operators of type $\left|\ \right\rangle\left\langle\ \right|$ (where
ket- and bra- state vectors need not to be Hermitian-conjugate to each other)
can not be directly performed by the Newton-Leibniz rule. Thus we invented an
innovative technique of integration within an ordered product (IWOP) of
operators that made the integration of non-commutative operators possible. The
core of IWOP technique is to arrange non-commutable quantum operators within
an ordered product (say, normal ordering) in a way that they become
commutable, in this sense the gap between q-numbers and c-numbers is
”narrowed”. However, the nature of operators which which are within : : is not
changed, they are still q-numbers, not c-numbers. After the integration over
c-numbers within ordered product is performed, we can get rid of the normal
ordering symbol after putting the integration result in normal ordering.
[37].The IWOP technique thus bridges this mathematical gap between classical
mechanics and quantum mechanics, and further reveals the beauty and elegance
of Dirac’s symbolic method and transformation theory. This technique develops
symbolic method significantly, i. e. makes Dirac’s representation theory and
the transformation theory more plentiful, and consequently to be better
understood. The beauty and elegance of Dirac’s symbolic method are further
revealed. Various applications of the IWOP technique, including constructing
the entangled state, developing the nonlinear coherent state theory, Wigner
function theory, etc. are found; many new unitary operators and operator-
identities as well as new quantum mechanical representations can be derived
too, which are partly summarized in the Review Articles [12].

We begin with listing some properties of normal product of operators which
means all the bosonic creation operators $a^{\dagger}$ are standing on the
left of annihilation operators $a$ in a monomial of $a^{\dagger}$ and $a$.

1\. The order of Bose operators $a$ and $a^{\dagger}$ within a normally
ordered product can be permuted. That is to say, even though
$\left[a,a^{\dagger}\right]=1$, we can have $\colon aa^{\dagger}\colon=\colon
a^{\dagger}a\colon=a^{\dagger}a,$ where $:$ $:$ denotes normal ordering.

2\. $c$-numbers can be taken out of the symbol $:$ $\colon$ as one wishes.

3\. The symbol $:$ : which is within another symbol $:$ $\colon$ can be
deleted.

4\. The vacuum projection operator $|0\rangle\langle 0|$ has the normal
product form

$|0\rangle\langle 0|=\colon e^{-a^{\dagger}a}\colon.$ (13)

5\. A normally ordered product can be integrated or differentiated with
respect to a $c$-number provided the integration is convergent.

###  3.2 The IWOP technique for deriving normally ordered Gaussian form of the
completeness relations of fundamental quantum mechanical representations

As an application of IWOP, (in the following, unless particularly mentioned,
we take $\hbar=\omega=m=1$ for convenience.) Using the Fock representation of
the coordinate eigenvector $Q|q\rangle=q|q\rangle,$
($Q=(a+a^{\dagger})/\sqrt{2}$)

$|q\rangle=\pi^{-1/4}e^{-\frac{q^{2}}{2}+\sqrt{2}qa^{\dagger}-\frac{a^{\dagger
2}}{2}}|0\rangle,$ (14)

we perform the integration below

$\displaystyle S_{1}$
$\displaystyle\equiv\int_{-\infty}^{\infty}\frac{dq}{\sqrt{\mu}}|\frac{q}{\mu}\rangle\langle
q|$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dq}{\sqrt{\pi\mu}}e^{-\frac{q^{2}}{2\mu^{2}}+\sqrt{2}\frac{q}{\mu}a^{\dagger}-\frac{a^{\dagger
2}}{2}}|0\rangle\langle 0|e^{-\frac{q^{2}}{2}+\sqrt{2}qa-\frac{a^{2}}{2}}.$
(15)

Substituting (13) into (15) we see

$S_{1}=\int_{-\infty}^{\infty}\frac{dq}{\sqrt{\pi\mu}}e^{-\frac{q^{2}}{2\mu^{2}}+\sqrt{2}\frac{q}{\mu}a^{\dagger}-\frac{a^{\dagger
2}}{2}}\colon e^{-a^{\dagger}a}\colon
e^{-\frac{q^{2}}{2}+\sqrt{2}qa-\frac{a^{2}}{2}}.$ (16)

Note that on the left of $\colon e^{-a^{+}a}\colon$ are all creation
operators, while on its right are all annihilation operators, so the whole
integral is in normal ordering, thus using property 1 we have

$S_{1}=\int_{-\infty}^{\infty}\frac{dq}{\sqrt{\pi\mu}}\colon
e^{-\frac{q^{2}}{2}(1+\frac{1}{\mu^{2}})+\sqrt{2}q(\frac{a^{\dagger}}{\mu}+a)-\frac{1}{2}(a+a^{\dagger})^{2}}\colon.$
(17)

As $a$ commutes with $a^{\dagger\text{ }}$ within $:$ $:$, so
$a^{\dagger\text{ }}$and $a$ can be considered as if they were parameters
while the integration is performing. Therefore, by setting $\mu=e^{\lambda}$,
sech$\lambda=\frac{2\mu}{1+\mu^{2}},$
tanh$\lambda=\frac{\mu^{2}+1}{\mu^{2}-1},$ we are able to perform the
integration and obtain

$\displaystyle S_{1}$
$\displaystyle=\sqrt{\frac{2\mu}{1+\mu^{2}}}\colon\exp\left\\{\frac{\left(\frac{a^{\dagger}}{\mu}+a\right)^{2}}{1+\frac{1}{\mu^{2}}}-\frac{1}{2}\left(a+a^{\dagger}\right)^{2}\right\\}\colon$
$\displaystyle=\left(\operatorname*{sech}\lambda\right)^{1/2}e^{-\frac{a^{\dagger
2}}{2}\tanh\lambda}\colon
e^{\left(\operatorname*{sech}\lambda-1\right)a^{\dagger}a}\colon
e^{\frac{a^{2}}{2}\tanh\lambda},$ (18)

which is just the single-mode squeezing operator in normal ordering appearing
in many references. It is worth mentioning that we have not used the SU(1,1)
Lie algebra method in the derivation. The integration automatically arranges
the squeezing operator in normal ordering. Using

$\displaystyle e^{\lambda a^{\dagger}a}$
$\displaystyle=\sum_{n=0}^{\infty}e^{\lambda n}|n\rangle\langle
n|=\sum_{n=0}^{\infty}e^{\lambda n}\frac{a^{\dagger n}}{n!}\colon
e^{-a^{\dagger}a}\colon a^{n}$
$\displaystyle=\colon\exp[\left(e^{\lambda}-1\right)a^{\dagger}a]\colon,$ (19)

Eq. (18) becomes

$\int_{-\infty}^{\infty}\frac{dq}{\sqrt{\mu}}|\frac{q}{\mu}\rangle\langle
q|=e^{-\frac{a^{+2}}{2}\tanh\lambda}e^{(a^{+}a+\frac{1}{2})\ln\operatorname*{sech}\lambda}e^{\frac{a^{2}}{2}\tanh\lambda}.$
(20)

This shows the classical dilation $q\rightarrow\frac{q}{\mu}$ maps into the
normally ordered squeezing operator manifestly. It also exhibits that the
fundamental representation theory can be formulated in not so abstract way, as
we can now directly perform the integral over ket-bra projection operators.
Moreover, the IWOP technique can be employed to perform many complicated
integrations for ket-bra projection operators.

There is a deep ditch between quantum mechanical operators ($q$-numbers)
theory and classical numbers ($c$-numbers) theory. The IWOP technique arranges
non-commutable operators within an ordered product symbol in a way that they
become commutable, in this sense the ‘ditch’ between $q$-numbers and
$c$-numbers is ”shoaled”. However, the nature of operators are not changed,
they are still $q-$numbers, not $c$-numbers. After the integration over
$c$-numbers within ordered product is performed, we can finally get rid of the
normal ordering symbol by using (19).

When $\mu=1,$ Eq. (20) becomes

$\displaystyle\int_{-\infty}^{\infty}dq|q\rangle\langle q|$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dq}{\sqrt{\pi}}\colon
e^{-q^{2}+2q(\frac{a+a^{\dagger}}{\sqrt{2}})-\frac{1}{2}(a+a^{\dagger})^{2}}\colon$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dq}{\sqrt{\pi}}\colon
e^{-\left(q-Q\right)^{2}}\colon=1,\;\;$ (21)

a real simple Gaussian integration! This immediately leads us to put the
completeness relation of the momentum representation into the normally ordered
Gaussian form

$\int_{-\infty}^{\infty}dp|p\rangle\langle
p|=\int_{-\infty}^{\infty}\frac{dp}{\sqrt{\pi}}\colon
e^{-\left(p-P\right)^{2}}\colon=1,$ (22)

where $P=\left(a-a^{\dagger}\right)/(i\sqrt{2}),$ and $\left|p\right\rangle$
is the momentum eigenvector $P\left|p\right\rangle=p\left|p\right\rangle$,

$\left|p\right\rangle=\pi^{-\frac{1}{4}}\exp\left[-\frac{1}{2}p^{2}+i\sqrt{2}pa^{\dagger}+\frac{1}{2}a^{\dagger
2}\right]\left|0\right\rangle.$ (23)

In addition, we should notice that $\left|q\right\rangle$ and
$\left|p\right\rangle$ are related by the Fourier transform (FT), i.e.
$\left\langle
p\right|\left.q\right\rangle=\frac{1}{\sqrt{2\pi}}\exp\left(-iqp\right),$ the
integral kernel of the Fraunhofer diffraction formula in 1-dimensional is such
a FT, so FT in classical optics has its correspondence in quantum mechanical
representations’ transform. This enlightens us that in order to find more
general analogy between unitary operators in quantum optics and
transformations in classical optics we should construct new representations
for quantum optics theory, and these are the bi-partite entangled state and
many-particle entangled state. These ideal states can be implemented by
optical devices and optical network [38]. In the following we focus on the bi-
partite entangled state.

###  3.3 Single-mode Wigner operator

When we combine (21) and (22) we can obtain

$\pi^{-1}\colon
e^{-\left(q-Q\right)^{2}-\left(p-P\right)^{2}}\colon\equiv\Delta\left(q,p\right),$
(24)

which is just the normally ordered Wigner operator since its marginal
integration gives $|q\rangle\langle q|$ and $|p\rangle\langle p|$
respectively, i.e.,

$\displaystyle\int_{-\infty}^{\infty}dq\Delta\left(q,p\right)$
$\displaystyle=\frac{1}{\sqrt{\pi}}\colon
e^{-\left(p-P\right)^{2}}\colon=|p\rangle\langle p|,$ (25)
$\displaystyle\int_{-\infty}^{\infty}dp\Delta\left(q,p\right)$
$\displaystyle=\frac{1}{\sqrt{\pi}}\colon
e^{-\left(q-Q\right)^{2}}\colon=|q\rangle\langle q|.$ (26)

Thus the Wigner function of quantum state $\rho$ can be calculated as
$W(q,p)=$Tr$[\rho\Delta\left(q,p\right)]$. On the other hand, the Wigner
operator (24) can be recast into the coherent state representation,

$\Delta\left(q,p\right)\rightarrow\Delta\left(\alpha,\alpha^{\ast}\right)=\int\frac{d^{2}z}{\pi}\left|\alpha+z\right\rangle\left\langle\alpha-z\right|e^{\alpha
z^{\ast}-\alpha^{\ast}z},$ (27)

where $\left|z\right\rangle$ is a coherent state. In fact, using the IWOP
technique we can obtain

$\displaystyle\Delta\left(\alpha,\alpha^{\ast}\right)$
$\displaystyle=\int\frac{d^{2}z}{\pi}\colon\exp\\{-\left|z\right|^{2}+\left(\alpha+z\right)a^{{\dagger}}+\left(\alpha^{\ast}-z^{\ast}\right)a$
$\displaystyle+\alpha
z^{\ast}-\alpha^{\ast}z-\left|\alpha\right|^{2}\\}\colon$
$\displaystyle=\frac{1}{\pi}\colon\exp\left\\{-2\left(a-\alpha\right)\left(a^{{\dagger}}-\alpha^{\ast}\right)\right\\}\colon,$
(28)

which is the same as (24).

###  3.4 Entangled state $\left|\eta\right\rangle$ and its Fourier transform
in complex form

The concept of quantum entanglement was first employed by Einstein, Rosen and
Poldosky (EPR) to challenge that quantum mechanics is incomplete when they
observed that two particles’ relative position $Q_{1}-Q_{2}$ and the total
momentum $P_{1}+P_{2}$ are commutable. Hinted by EPR, the bipartite entangled
state $\left|\eta\right\rangle$ is introduced as [39, 40]

$\left|\eta\right\rangle=\exp\left[-\frac{1}{2}\left|\eta\right|^{2}+\eta
a_{1}^{\dagger}-\eta^{\ast}a_{2}^{\dagger}+a_{1}^{\dagger}a_{2}^{\dagger}\right]\left|00\right\rangle.$
(29)

$\left|\eta=\eta_{1}+\mathtt{i}\eta_{2}\right\rangle$ is the common eigenstate
of relative coordinate $Q_{1}-Q_{2}$ and the total momentum $P_{1}+P_{2}$,

$\left(Q_{1}-Q_{2}\right)\left|\eta\right\rangle=\sqrt{2}\eta_{1}\left|\eta\right\rangle,\text{
}\,\text{\
}\left(P_{1}+P_{2}\right)\left|\eta\right\rangle=\sqrt{2}\eta_{2}\left|\eta\right\rangle,$
(30)

where $Q_{i}=(a_{j}+a_{j}^{\dagger})/\sqrt{2},\
P_{j}=(a_{j}-a_{j}^{\dagger})/(\mathtt{i}\sqrt{2}),$ $j=1,2.$ Using the IWOP
technique, we can immediately prove that $\left|\eta\right\rangle$ possesses
the completeness relation

$\int\frac{d^{2}\eta}{\pi}\left|\eta\right\rangle\left\langle\eta\right|=\int\frac{d^{2}{\eta}}{\pi}\colon
e^{-\left[\eta^{\ast}-(a_{1}^{\dagger}-a_{2})\right]\left[\eta-(a_{1}-a_{2}^{\dagger})\right]}\colon=1,\,d^{2}\eta=d\eta_{1}d\eta_{2},$
(31)

and orthonormal relation

$\left\langle\eta\right|\left.\eta^{\prime}\right\rangle=\pi\delta(\eta_{1}-\eta_{1}^{\prime})\delta(\eta_{2}-\eta_{2}^{\prime}).$
(32)

The Schmidt decomposition of $\left|\eta\right\rangle$ is

$\left|\eta\right\rangle=e^{-i\eta_{2}\eta_{1}}\int_{-\infty}^{\infty}dx\left|q\right\rangle_{1}\otimes\left|q-\sqrt{2}\eta_{1}\right\rangle_{2}e^{i\sqrt{2}\eta_{2}x},$
(33)

The $\left|\eta\right\rangle$ state can also be Schmidt-decomposed in momentum
eigenvector space as

$\left|\eta\right\rangle=e^{i\eta_{1}\eta_{2}}\int_{-\infty}^{\infty}dp\left|p\right\rangle_{1}\otimes\left|\sqrt{2}\eta_{2}-p\right\rangle_{2}e^{-i\sqrt{2}\eta_{1}p}.$
(34)

The $\left|\eta\right\rangle$ is physically appealing in quantum optics
theory, because the two-mode squeezing operator has its natural representation
on $\left\langle\eta\right|$ basis [40]

${\displaystyle\int}\frac{d^{2}\eta}{\pi\mu}\left|\eta/\mu\right\rangle\left\langle\eta\right|=e^{a_{1}^{{}^{\dagger}}a_{2}^{{}^{\dagger}}\tanh\lambda}e^{(a_{1}^{{}^{\dagger}}a_{1}+a_{2}^{{}^{\dagger}}a_{2}+1)\ln\operatorname*{sech}\lambda}e^{-a_{1}a_{2}\tanh\lambda},\;\mu=e^{\lambda},$
(35)

The proof of (35) is proceeded by virtue of the IWOP technique

$\displaystyle{\displaystyle\int}\frac{d^{2}\eta}{\pi\mu}\left|\eta/\mu\right\rangle\left\langle\eta\right|$
$\displaystyle={\displaystyle\int}\frac{d^{2}\eta}{\pi\mu}\colon\exp\left\\{-\frac{|\eta|^{2}}{2}\left(1+\frac{1}{\mu^{2}}\right)+\eta\left(\frac{a_{1}^{{}^{\dagger}}}{\mu}-a_{2}\right)\right.$
$\displaystyle+\left.\eta^{\ast}\left(a_{1}-\frac{a_{2}^{{}^{\dagger}}}{\mu}\right)+a_{1}^{\dagger}a_{2}^{{}^{\dagger}}+a_{1}a_{2}-a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}\right\\}$
$\displaystyle=\frac{2\mu}{1+\mu^{2}}\colon\exp\left\\{\frac{\mu^{2}}{1+\mu^{2}}\left(\frac{a_{1}^{{}^{\dagger}}}{\mu}-a_{2}\right)\left(a_{1}-\frac{a_{2}^{{}^{\dagger}}}{\mu}\right)-\left(a_{1}-a_{2}^{{}^{\dagger}}\right)\left(a_{1}^{{}^{\dagger}}-a_{2}\right)\right\\}\colon$
$\displaystyle=e^{a_{1}^{{}^{\dagger}}a_{2}^{{}^{\dagger}}\tanh\lambda}e^{(a_{1}^{{}^{\dagger}}a_{1}+a_{2}^{{}^{\dagger}}a_{2}+1)\ln\operatorname*{sech}\lambda}e^{-a_{1}a_{2}\tanh\lambda}\equiv
S_{2},$ (36)

so the necessity of introducing $\left|\eta\right\rangle$ into quantum optics
is clear. $S_{2}$ squeezes $\left|\eta\right\rangle$ in the manifest way

$S_{2}\left|\eta\right\rangle=\frac{1}{\mu}\left|\eta/\mu\right\rangle,\text{
\ }\mu=e^{\lambda}\ ,$ (37)

and the two-mode squeezed state itself is an entangled state which entangles
the idle mode and signal mode as an outcome of a parametric-down conversion
process [41].

We can also introduce the conjugate state of $\left|\eta\right\rangle$ [42],

$\left|\xi\right\rangle=\exp\left[-\frac{1}{2}\left|\xi\right|^{2}+\xi
a_{1}^{\dagger}+\xi^{\ast}a_{2}^{\dagger}-a_{1}^{\dagger}a_{2}^{\dagger}\right]\left|00\right\rangle,\text{
}\xi=\xi_{1}+i\xi_{2},$ (38)

which obeys the eigen-equations

$\left(Q_{1}+Q_{2}\right)\left|\xi\right\rangle=\sqrt{2}\xi_{1}\left|\xi\right\rangle,\,\left(P_{1}-P_{2}\right)\left|\xi\right\rangle=\sqrt{2}\xi_{2}\left|\xi\right\rangle.$
(39)

Because
$\left[\left(Q_{1}-Q_{2}\right),\left(P_{1}-P_{2}\right)\right]=2\mathtt{i},$
so we name the conjugacy between $\left|\xi\right\rangle$ and
$\left|\eta\right\rangle.$ The completeness and orthonormal relations of
$\left|\xi\right\rangle$ are

$\displaystyle\int\frac{d^{2}\xi}{\pi}\left|\xi\right\rangle\left\langle\xi\right|$
$\displaystyle=\int\frac{d^{2}{\xi}}{\pi}\colon
e^{-\left[\xi^{\ast}-(a_{1}^{\dagger}+a_{2})\right]\left[\xi-(a_{1}+a_{2}^{\dagger})\right]}\colon=1,$
(40) $\displaystyle\left\langle\xi\right|\left.\xi^{\prime}\right\rangle$
$\displaystyle=\pi\delta(\xi_{1}-\xi_{1}^{\prime})\delta(\xi_{2}-\xi_{2}^{\prime}),\text{
}d^{2}\xi=d\xi_{1}d\xi_{2},$ (41)

respectively. $\left|\eta\right\rangle$ and $\left|\xi\right\rangle$ can be
related to each other by

$\left\langle\eta|\xi\right\rangle=\frac{1}{2}\exp\left(\frac{\xi\eta^{\ast}-\xi^{\ast}\eta}{2}\right),$
(42)

since $\xi^{\ast}\eta-\xi\eta^{\ast}$ is a pure imaginary number, Eq. (42) is
the Fourier transform kernel in complex form (or named entangled Fourier
transform, this concept should also be extended to multipartite entangled
states.) It will be shown in later sections that departing from entangled
states $\left|\eta\right\rangle$ and $\left|\xi\right\rangle$ and the
generalized Fresnel operator a new entangled Fresnel transforms in classical
optics can be found.

###  3.5 Two-mode Wigner operator in the $\left|\eta\right\rangle$
representation

Combining (31) and (40) we can construct the following operator

$\displaystyle\frac{1}{\pi^{2}}\colon
e^{-\left[\sigma^{\ast}-(a_{1}^{\dagger}-a_{2})\right]\left[\sigma-(a_{1}-a_{2}^{\dagger})\right]-\left[\gamma^{\ast}-(a_{1}^{\dagger}+a_{2})\right]\left[\gamma-(a_{1}+a_{2}^{\dagger})\right]}\colon$
$\displaystyle=\Delta\left(\alpha,\alpha^{\ast}\right)\otimes\Delta\left(\beta,\beta^{\ast}\right)\equiv\Delta\left(\sigma,\gamma\right),$
(43)

where

$\sigma=\alpha-\beta^{\ast},\;\gamma=\alpha+\beta^{\ast}.$ (44)

Eq. (43) is just equal to the direct product of two single-mode Wigner
operators. It is convenient to express the Wigner operator in the
$\left|\eta\right\rangle$ representation as [43]

$\Delta\left(\sigma,\gamma\right)=\int\frac{d^{2}\eta}{\pi^{3}}\left|\sigma-\eta\right\rangle\left\langle\sigma+\eta\right|e^{\eta\gamma^{\ast}-\eta^{\ast}\gamma}.$
(45)

For two-mode correlated system, it prefers to using
$\Delta\left(\sigma,\gamma\right)$ to calculate quantum states’ Wigner
function. For example, noticing
$\left\langle\eta\right|\left.00\right\rangle=\exp\\{-\left|\eta\right|^{2}/2\\},$
the two-mode squeezed states’ Wigner function is

$\displaystyle\left\langle
00\right|S_{2}^{{\dagger}}\left(\mu\right)\Delta\left(\sigma,\gamma\right)S_{2}\left(\mu\right)\left|00\right\rangle$
$\displaystyle=\left\langle
00\right|\mu^{2}\int\frac{d^{2}\eta}{\pi^{3}}\left|\mu\left(\sigma-\eta\right)\right\rangle\left\langle\mu\left(\sigma+\eta\right)\right|e^{\eta\gamma^{\ast}-\eta^{\ast}\gamma}\left|00\right\rangle$
$\displaystyle=\pi^{-2}\exp\left[-\mu^{2}\left|\sigma\right|^{2}-\left|\gamma\right|^{2}/\mu^{2}\right].$
(46)

##  4 Two deduced entangled state representations and Hankel transform

###  4.1 Deduced entangled states

Starting from the entangled state $\left|\eta=re^{i\theta}\right\rangle$ and
introducing an integer $m$, we can deduce new states [44],

$\left|m,r\right\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\left|\eta=re^{i\theta}\right\rangle
e^{-im\theta},$ (47)

which is worth of paying attention because when we operate the number-
difference operator,

$D\equiv a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}$ (48)

on $\left|\eta\right\rangle,$ using Eq.(29) we see

$D\left|\eta\right\rangle=\left(\eta
a_{1}^{{}^{\dagger}}+\eta^{\ast}a_{2}^{{}^{\dagger}}\right)\left|\eta\right\rangle=-i\frac{\partial}{\partial\theta}\left|\eta\right\rangle,\;\eta=|\eta|e^{i\theta},$
(49)

so the number-difference operator corresponds to a differential operation
$i\frac{\partial}{\partial\theta}$ in the
$\left\langle\eta\right|\;$representation, this is a remarkable property of
$\left\langle\eta\right|$. It then follows

$D\left|m,r\right\rangle=\int_{0}^{2\pi}\frac{d\theta}{2\pi}e^{-im\theta}\left(-i\frac{\partial}{\partial\theta}\left|\eta=re^{i\theta}\right\rangle\right)=m\left|m,r\right\rangle.$
(50)

On the other hand, by defining

$K\equiv(a_{1}-a_{2}^{\dagger})(a_{1}^{\dagger}-a_{2}),$ (51)

we see $\left[D,K\right]=0,$ and $\left|m,r\right\rangle$ is its eigenstate,

$\ K\left|m,r\right\rangle=r^{2}\left|m,r\right\rangle,$ (52)

where $K$ is named correlated-amplitude operator since
$K\left|\eta\right\rangle=|\eta|^{2}\left|\eta\right\rangle.$ Thus we name
$\left|m,r\right\rangle$ correlated-amplitude—number-difference entangled
states. It is not difficult to prove completeness and orthonormal property of
$\left|m,r\right\rangle$,

$\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{2}\right)\left|m,r\right\rangle\left\langle
m,r\right|=1,\;$ (53) $\left\langle
m,r\right|\left.m^{\prime},r^{\prime}\right\rangle=\delta_{m,m^{\prime}}\frac{1}{2r}\delta\left(r-r^{\prime}\right).$
(54)

On the other hand, from $\left|\xi\right\rangle$ we can derive another state

$\left|s,r^{\prime}\right\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}d\varphi\left|\xi=r^{\prime}e^{i\varphi}\right\rangle
e^{-is\varphi},$ (55)

which satisfies

$D\left|\xi\right\rangle=\left(a_{1}^{\dagger}\xi-
a_{2}^{\dagger}\xi^{\ast}\right)\left|\xi\right\rangle=-i\frac{\partial}{\partial\varphi}\left|\xi=r^{\prime}e^{i\varphi}\right\rangle.$
(56)

So $D$ in $\left\langle\xi=r^{\prime}e^{i\varphi}\right|$ representation is
equal to $i\frac{\partial}{\partial\varphi}$. Consequently,

$D\left|s,r^{\prime}\right\rangle=\int_{0}^{2\pi}\frac{d\theta}{2\pi}e^{-is\theta}\left(-i\frac{\partial}{\partial\theta}\left|\xi=r^{\prime}e^{i\theta}\right\rangle\right)=s\left|s,r^{\prime}\right\rangle.$
(57)

Note $\left[D,(a_{1}^{\dagger}+a_{2})(a_{1}+a_{2}^{\dagger})\right]=0$ and

$(a_{1}^{\dagger}+a_{2})(a_{1}+a_{2}^{\dagger})\left|s,r^{\prime}\right\rangle=r^{\prime
2}\left|s,r^{\prime}\right\rangle.$ (58)

$\left|s,r^{\prime}\right\rangle$ is qualified to be a new representation
since

$\sum_{s=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{\prime
2}\right)\left|s,r^{\prime}\right\rangle\left\langle
s,r^{\prime}\right|=1,\text{ }\left\langle
s,r^{\prime}\right|\left.s^{\prime},r^{\prime\prime}\right\rangle=\delta_{s,s^{\prime}}\frac{1}{2r^{\prime}}\delta\left(r^{\prime}-r^{\prime\prime}\right).$
(59)

###  4.2 Hankel transform between two deduced entangled state representations

Since $\left|\xi\right\rangle$ and $\left|\eta\right\rangle$ are mutual
conjugate, $\left|s,r^{\prime}\right\rangle$ is the conjugate state of
$\left|m,r\right\rangle$. From the definition of $\left|m,r\right\rangle$ and
$\left|s,r^{\prime}\right\rangle$ and (42) we calculate the overlap [21]

$\displaystyle\left\langle s,r^{\prime}\right|\left.q,r\right\rangle$
$\displaystyle=\frac{1}{4\pi^{2}}\int_{0}^{2\pi}d\varphi
e^{is\varphi}\left\langle\xi=r^{\prime}e^{i\varphi}\right|\int_{0}^{2\pi}d\theta\left|\eta=re^{i\theta}\right\rangle
e^{-im\theta}$
$\displaystyle=\frac{1}{8\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}e^{is\varphi-
im\theta}\exp\left[irr^{\prime}\sin\left(\theta-\varphi\right)\right]d\theta
d\varphi$
$\displaystyle=\frac{1}{8\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}e^{is\varphi-
im\theta}\sum_{l=-\infty}^{\infty}J_{l}\left(rr^{\prime}\right)e^{il\left(\theta-\varphi\right)}$
$\displaystyle=\frac{1}{2}\sum_{l=-\infty}^{\infty}\delta_{l,m}\delta_{l,s}J_{l}\left(rr^{\prime}\right)=\frac{1}{2}\delta_{s,m}J_{s}\left(rr^{\prime}\right),$
(60)

where we have identified the generating function of the s-order Bessel
function $J_{l},$

$e^{ix\sin t}=\sum_{l=-\infty}^{\infty}J_{l}\left(x\right)e^{ilt},\;$ (61)

and

$J_{l}\left(x\right)=\sum_{k=0}^{\infty}\frac{\left(-1\right)^{l}}{k!\left(l+k\right)!}\left(\frac{x}{2}\right)^{l+2k}.$
(62)

Eq. (60) is remarkable, because $J_{s}\left(rr^{\prime}\right)$ is just the
integral kernel of Hankel transform. In fact, if we define

$\left\langle m,r\right|\left.g\right\rangle\equiv g\left(m,r\right),\text{
}\left\langle
s,r^{\prime}\right|\left.g\right\rangle\equiv\mathcal{G}\left(s,r^{\prime}\right),$
(63)

and use (53) as well as (60), we obtain

$\displaystyle\mathcal{G}\left(s,r^{\prime}\right)$
$\displaystyle=\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{2}\right)\left\langle
s,r^{\prime}\right|\left.m,r\right\rangle\left\langle
m,r\right|\left.g\right\rangle$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}d\left(r^{2}\right)J_{s}\left(rr^{\prime}\right)g\left(s,r\right)\equiv\mathcal{H}\left[g\left(s,r\right)\right],$
(64)

which is just the Hankel transform of $g\left(m,r\right)$ (or it can be
regarded as a simplified form of the Collins formula in cylindrical
coordinate, see (10)). The inverse transform of (64) is

$\displaystyle g\left(m,r\right)$ $\displaystyle=\left\langle
m,\mathfrak{r}\right|\sum_{s=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{\prime
2}\right)\left|s,r^{\prime}\right\rangle\left\langle
s,r^{\prime}\right|\left.g\right\rangle$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}d\left(r^{\prime
2}\right)J_{q}\left(rr^{\prime}\right)\mathcal{G}\left(m,r^{\prime}\right)\equiv\mathcal{H}^{-1}\left[\mathcal{G}\left(m,r^{\prime}\right)\right].$
(65)

Now we know that the quantum optical image of classical Hankel transform just
corresponds to the representation transformation between two mutually
conjugate entangled states $\left\langle s,r^{\prime}\right|$ and
$\left|m,r\right\rangle,$ this is like the case that the Fourier transform
kernel is just the matrix element between the coordinate state and the
momentum state, a wonderful result unnoticed before. Therefore the bipartite
entangled state representations’ transforms, which can lead us to the Hankel
transform, was proposed first in classical optics, can find their way back in
quantum optics.

###  4.3 Quantum optical version of classical circular harmonic correlation

From Eq.(47) we can see that its reciprocal relation is the circular harmonic
expansion,

$\left|\eta=re^{i\theta}\right\rangle=\sum_{m=-\infty}^{\infty}\left|m,r\right\rangle
e^{im\theta},$ (66)

or correlated-amplitude—number-difference entangled state
$\left|m,r\right\rangle$ can be considered as circular harmonic decomposition
of $\left|\eta=re^{i\theta}\right\rangle.$ Let $g\left(r,\theta\right)$, a
general 2-dimensional function expressed in polar coordinates, be periodic in
the variable $\theta,$ it can be looked as the wavefunction of the state
vector $\left|g\right\rangle$ in the $\left\langle\eta=re^{i\theta}\right|$
representation

$g\left(r,\theta\right)=\left\langle\eta=re^{i\theta}\right.\left|g\right\rangle,\text{
}\text{\ }$ (67)

using (66) we have

$g\left(r,\theta\right)=\sum_{m=-\infty}^{\infty}g_{m}\left(r\right)e^{-im\theta},\text{
\ }g_{m}\left(r\right)=\left\langle m,r\right.\left|g\right\rangle,$ (68)

$g_{m}\left(r\right)$ is the wavefunction of $\left|g\right\rangle$ in
$\left\langle m,r\right|$ representation. By using (53) and noticing that It
then follows from (49)

$e^{-i\alpha(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2})}\left|\eta=re^{i\theta}\right\rangle=e^{-\alpha\frac{\partial}{\partial\theta}}\left|\eta=re^{i\theta}\right\rangle=\left|\eta=re^{i\left(\theta-\alpha\right)}\right\rangle,$
(69)

so
$e^{-i\alpha\left(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2}\right)}$
behaves a rotation operator in $\left|\eta\right\rangle$ representation, we
see that the expectation value of
$e^{-i\alpha\left(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2}\right)}$
in $\left|g\right\rangle$ is

$\displaystyle\pi\left\langle
g\right|e^{-i\alpha(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{{\dagger}}}a_{2})}\left|g\right\rangle$
$\displaystyle=\pi\left\langle
g\right|\int\frac{d^{2}\eta}{\pi}e^{-i\alpha(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2})}\left|\eta\right\rangle\left\langle\eta\right.\left|g\right\rangle$
$\displaystyle=\int_{0}^{\infty}rdrd\theta\int_{0}^{2\pi}\left\langle
g\right.\left|\eta^{\prime}=re^{i\left(\theta-\alpha\right)}\right\rangle\left\langle\eta=re^{i\theta}\right.\left|g\right\rangle$
$\displaystyle=\int_{0}^{\infty}rdr\int_{0}^{2\pi}g^{\ast}\left(r,\theta-\alpha\right)g\left(r,\theta\right)d\theta\equiv
R_{\alpha},$ (70)

which is just the cross-correlation between $g\left(r,\theta\right)$ and an
angularly rotated version of the same function,
$g^{\ast}\left(r,\theta-\alpha\right)$. On the other hand, using (53) we have

$\left|g\right\rangle=\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{2}\right)\left|m,r\right\rangle\left\langle
m,r\right|\left.g\right\rangle=\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{2}\right)\left|m,r\right\rangle
g_{m}\left(r\right).$ (71)

Substituting (71) into (70) and using the eigenvector equation (50) as well as
(54) we obtain

$\displaystyle R_{\alpha}$
$\displaystyle=\pi\sum_{m^{\prime}=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{\prime
2}\right)\left\langle
m^{\prime},r^{\prime}\right|g_{m^{\prime}}^{\ast}\left(r^{\prime}\right)e^{-i\alpha\left(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2}\right)}\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{2}\right)\left|m,r\right\rangle
g_{m}\left(r\right)$
$\displaystyle=\pi\sum_{m^{\prime}=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{\prime
2}\right)g_{m^{\prime}}^{\ast}\left(r^{\prime}\right)e^{-im\alpha}\int_{0}^{\infty}d\left(r^{2}\right)g_{m}\left(r\right)\delta_{m,m^{\prime}}\frac{1}{2r}\delta\left(r-r^{\prime}\right)$
$\displaystyle=2\pi\sum_{m=-\infty}^{\infty}e^{-im\alpha}\int_{0}^{\infty}r|g_{m}\left(r\right)|^{2}dr,$
(72)

from which we see that each of the circular harmonic components of the
crosscorrelation undergoes a different phase shift $-m\alpha,$ so $R_{\alpha}$
is not rotation invariant. However, when we consider only one harmonic
component

$R_{\alpha,M}=2\pi
e^{-iM\alpha}\int_{0}^{\infty}r|g_{M}\left(r\right)|^{2}dr,$ (73)

is extracted digitally, then from the phase associated with this component it
is possible to determine the angular shift that one version of the object has
undergone. When an optical filter that is matched to $R_{\alpha,M}$ of a
particular object is constructed, then if that some object is entered as an
input to the system with any angular rotation, a correlation peak of strength
proportional to $\int_{0}^{\infty}r|g_{M}\left(r\right)|^{2}dr$ will be
produced, independent of rotation. Hence an optical correlator can be
constructed that will recognize that object independent of rotation [19].

So far we have studied the circular harmonic correlation in the context of
quantum optics, we have endowed the crosscorrelation $R_{\alpha}$ with a
definite quantum mechanical meaning, i.e. the overlap between $\left\langle
g\right|$ and the rotated state
$e^{i\alpha\left(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2}\right)}\left|g\right\rangle,$
in the entangled state representation.

Note that Fourier-based correlators is also very sensitive to magnification,
however, the magnitude of Mellin transform is independent of scale-size
changes in the input [19]. Now we examine when $\left|g\right\rangle$ is both
rotated and squeezed (by a two-mode squeezing operator
$S_{2}\left(\lambda\right)=\exp[\lambda(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2})])$,
then from (37) and (67) we have

$S_{2}\left(\lambda\right)\left|g\right\rangle=\int\frac{d^{2}\eta}{\pi\mu}\left|\eta/\mu\right\rangle\left\langle\eta\right.\left|g\right\rangle=\int\frac{d^{2}\eta}{\pi\mu}\left|\eta/\mu\right\rangle
g\left(r,\theta\right),$ (74)

it follows the overlap between $\left\langle g\right|$ and the state
$e^{-i\alpha(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2})}S\left(\lambda\right)\left|g\right\rangle,$

$\displaystyle W_{\alpha,\lambda}$ $\displaystyle\equiv\pi\left\langle
g\right|e^{-i\alpha\left(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2}\right)}S\left(\lambda\right)\left|g\right\rangle$
$\displaystyle=\left\langle
g\right|\int\frac{d^{2}\eta}{\mu}e^{-i\alpha\left(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2}\right)}\left|\eta/\mu\right\rangle
g\left(r,\theta\right)$
$\displaystyle=\int_{0}^{\infty}\frac{rdr}{\mu}\int_{0}^{2\pi}\left\langle
g\right.\left|\eta^{\prime}=e^{i\left(\theta-\alpha\right)}r/\mu\right\rangle
g\left(r,\theta\right)d\theta$
$\displaystyle=\int_{0}^{\infty}\frac{rdr}{\mu}\int_{0}^{2\pi}g^{\ast}\left(r/\mu,\theta-\alpha\right)g\left(r,\theta\right)d\theta,$
(75)

which corresponds to the crosscorrelation arising from combination of
squeezing and rotation (joint transform correlator). On the other hand, from
(47) and (37) we see

$S_{2}\left(\lambda\right)\left|m,r\right\rangle=\frac{1}{2\pi\mu}\int_{0}^{2\pi}d\theta\left|\eta=\frac{r}{\mu}e^{i\theta}\right\rangle
e^{-im\theta}=\frac{1}{\mu}\left|m,\frac{r}{\mu}\right\rangle,$ (76)

and therefore

$\displaystyle W_{\alpha,\lambda}$
$\displaystyle=\frac{\pi}{\mu}\sum_{m^{\prime}=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{\prime
2}\right)\left\langle
m^{\prime},r^{\prime}\right|g_{m^{\prime}}^{\ast}\left(r^{\prime}\right)e^{-i\alpha\left(a_{1}^{{}^{\dagger}}a_{1}-a_{2}^{{}^{\dagger}}a_{2}\right)}\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{2}\right)\left|m,\frac{r}{\mu}\right\rangle
g_{m}\left(r\right)$
$\displaystyle=\frac{\pi}{\mu}\sum_{m^{\prime}=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}d\left(r^{\prime
2}\right)g_{m^{\prime}}^{\ast}\left(r^{\prime}\right)e^{-im\alpha}\int_{0}^{\infty}d\left(r^{2}\right)g_{m}\left(r\right)\delta_{m,m^{\prime}}\frac{1}{2r^{\prime}}\delta\left(\frac{r}{\mu}-r^{\prime}\right)$
$\displaystyle=\frac{2\pi}{\mu}\sum_{m=-\infty}^{\infty}e^{-im\alpha}\int_{0}^{\infty}rg_{m}\left(r\right)g_{m^{\prime}}^{\ast}\left(re^{-\lambda}\right)dr,\text{
\ }$ (77)

from which one can see that to achieve simultaneous scale and rotation
invariance, a two-dimensional object $g\left(r,\theta\right)$ should be
entered into the optical system in a distorted polar coordinate system, the
distortation arising from the fact that the radial coordinate is stretched by
a logarithmic transformation $\left(\lambda=-\ln\mu\right)$, which coincides
with Ref. [45]. The quantum optical version is thus established which is a new
tie connecting Fourier optics and quantum optics [46].

At the end of this section, using the two-variable Hermite polynomials’
definition [47]

$H_{m,n}\left(\xi,\xi^{\ast}\right)=\sum_{l=0}^{\min\left(m,n\right)}\frac{m!n!\left(-1\right)^{l}\xi^{m-l}\xi^{\ast
n-l}}{l!\left(m-l\right)!\left(n-l\right)!},$ (78)

which is quite different from the product of two single-variable Hermite
polynomials, and its generating function formula is

$\sum_{m,n=0}^{\infty}\frac{t^{m}t^{\prime
n}}{m!n!}H_{m,n}\left(\xi,\xi^{\ast}\right)=\exp\left[-tt^{\prime}+t\xi+t^{\prime}\xi^{\ast}\right],$
(79)

and noting
$H_{m,n}(\xi,\xi^{\ast})=e^{i(m-n)\varphi}H_{m,n}(r^{\prime},r^{\prime}),$ we
can directly perform the integral in (55) and derive the explicit form of
$\left|s,r^{\prime}\right\rangle,$

$\displaystyle\left|s,r^{\prime}\right\rangle$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}d\varphi\exp\\{-r^{\prime 2}/2+\xi
a_{1}^{\dagger}+\xi^{\ast}a_{2}^{\dagger}-is\varphi-
a_{1}^{\dagger}a_{2}^{\dagger}\\}\left|00\right\rangle$
$\displaystyle=\frac{1}{2\pi}e^{-r^{\prime
2}/2}\int_{0}^{2\pi}d\varphi\sum\limits_{m,n=0}^{\infty}\frac{a_{1}^{\dagger
m}a_{2}^{\dagger
n}}{m!n!}H_{m,n}(\xi,\xi^{\ast})e^{-is\varphi}\left|00\right\rangle$
$\displaystyle=\frac{1}{2\pi}e^{-r^{\prime
2}/2}\int_{0}^{2\pi}d\varphi\sum\limits_{m,n=0}^{\infty}\frac{1}{\sqrt{m!n!}}H_{m,n}(r^{\prime},r^{\prime})e^{i\varphi\left(m-n-s\right)}\left|m,n\right\rangle$
$\displaystyle=e^{-r^{\prime
2}/2}\sum\limits_{n=0}^{\infty}\frac{1}{\sqrt{\left(n+s\right)!n!}}H_{n+s,n}(r^{\prime},r^{\prime})\left|n+s,n\right\rangle,$
(80)

which is really an entangled state in two-mode Fock space. Eqs. (79) and (80)
will be often used in the following discussions.

In the following we concentrate on finding the generalized Fresnel operators
in both one- and two- mode cases with use of the IWOP technique.

##  5 Single-mode Fresnel operator as the image of the classical Optical
Fresnel Transform

In this section we shall mainly introduce so-called generalized Fresnel
operators (GFO) (in one-mode and two-mode cases both) [48] and some
appropriate quantum optical representations (e.g. coherent state
representation and entangled state representation) to manifestly link the
formalisms in quantum optics to those in classical optics. In so doing, we
find that the various transforms in classical optics are just the result of
generalized Fresnel operators inducing transforms on appropriate quantum state
vectors, i.e. classical optical Fresnel transforms have their counterpart in
quantum optics. Besides, we can study the important $ABCD$ rule obeyed by
Gaussian beam propagation (also the ray propagation in matrix optics) [49] in
the domain of quantum optics.

###  5.1 Single-mode GFO gained via coherent state method

For the coherent state $\left|z\right\rangle$ in quantum optics [1, 2]

$\left|z\right\rangle=\exp\left[za^{\dagger}-z^{\ast}a\right]|0\rangle\equiv\left|\left(\begin{array}[c]{c}z\\\
z^{\ast}\end{array}\right)\right\rangle,$ (81)

which is the eigenstate of annihilation operator $a,$
$a\left|z\right\rangle=z\left|z\right\rangle$, using the IWOP and (13), we can
put the over-completeness relation of $\left|z\right\rangle$ into normal
ordering

$\int\frac{d^{2}{z}}{\pi}\left|z\right\rangle\left\langle
z\right|=\int\frac{d^{2}{z}}{\pi}\colon
e^{-\left(z^{\ast}-a^{\dagger}\right)\left(z-a\right)}\colon=1.$ (82)

the canonical form of coherent state $\left|z\right\rangle$ is expressed as

$\left|z\right\rangle=\left|p,q\right\rangle=\exp\left[i\left(pQ-
qP\right)\right]|0\rangle\equiv\left|\left(\begin{array}[c]{c}q\\\
p\end{array}\right)\right\rangle,$ (83)

where $z=\left(q+\mathtt{i}p\right)/\sqrt{2}$. It follows that $\left\langle
p,q\right|Q\left|p,q\right\rangle=q,$ $\left\langle
p,q\right|P\left|p,q\right\rangle=p,$ this indicates that the states
$\left|p,q\right\rangle$ generate a canonical phase-space representation for a
state $\left|\Psi\right\rangle,$ $\Psi\left(p,q\right)=\left\langle
p,q\right|\left.\Psi\right\rangle.$ Thus the coherent state is a good
candidate for providing with classical phase-space description of quantum
systems. Remembering that the Fresnel transform’s parameters
$\left(A,B,C,D\right)$ are elements of a ray transfer matrix $M$ describing
optical systems, $M$ belongs to the unimodular symplectic group, and the
coherent state $\left|p,x\right\rangle$ is a good candidate for providing with
classical phase-space description of quantum systems, we naturally think of
that the symplectic transformation $\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)\left(\begin{array}[c]{c}q\\\ p\end{array}\right)$ in
classical phase space may mapping onto a generalized Fresnel operator in
Hilbert space through the coherent state basis. Thus we construct the
following ket-bra projection operator

$\iint\limits_{-\infty}^{\infty}dxdp\left|\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)\left(\begin{array}[c]{c}q\\\
p\end{array}\right)\right\rangle\left\langle\left(\begin{array}[c]{c}q\\\
p\end{array}\right)\right|$ (84)

as the GFO. In fact, using notation of $\left|z\right\rangle$ (coherent
state), and introducing complex numbers $s,r,$

$s=\frac{1}{2}\left[A+D-i\left(B-C\right)\right],\;r=-\frac{1}{2}\left[A-D+i\left(B+C\right)\right],\;|s|^{2}-|r|^{2}=1,$
(85)

from (83) we know

$\displaystyle\left|\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)\left(\begin{array}[c]{c}q\\\
p\end{array}\right)\right\rangle$
$\displaystyle=\left|\left(\begin{array}[c]{cc}s&-r\\\
-r^{\ast}&s^{\ast}\end{array}\right)\left(\begin{array}[c]{c}z\\\
z^{\ast}\end{array}\right)\right\rangle\equiv\left|sz-rz^{\ast}\right\rangle$
(94) $\displaystyle=\exp\left[-\frac{1}{2}|sz-rz^{\ast}|^{2}+(sz-
rz^{\ast})a^{\dagger}\right]|0\rangle,$ (95)

$\left(\begin{array}[c]{cc}s&-r\\\ -r^{\ast}&s^{\ast}\end{array}\right)$ is
still a symplectic group element, so (84) becomes [50]

$F_{1}\left(s,r\right)=\sqrt{s}\int\frac{d^{2}z}{\pi}\left|sz-
rz^{\ast}\right\rangle\left\langle z\right|,\;\;$ (96)

where the factor $\sqrt{s}$ is attached for anticipating the unitarity of the
operator $F_{1}.$ Eq. (84) tells us that $c$-number transform
$\left(q,p\right)\rightarrow\left(Aq+Bp,Cq+Dp\right)$ in coherent state basis
maps into $F_{1}\left(s,r\right)$. Now we prove $F_{1}\left(s,r\right)$ is
really the FO we want. Using the IWOP technique and Eq. (95) and (13) we can
perform the integral

$\displaystyle F_{1}\left(s,r\right)$
$\displaystyle=\sqrt{s}\int\frac{d^{2}z}{\pi}\colon\exp\left[-\left|s\right|^{2}\left|z\right|^{2}+sza^{\dagger}+z^{\ast}\left(a-ra^{\dagger}\right)+\frac{r^{\ast}s}{2}z^{2}+\frac{rs^{\ast}}{2}z^{\ast
2}-a^{\dagger}a\right]\colon$
$\displaystyle=\frac{1}{\sqrt{s^{\ast}}}\exp\left(-\frac{r}{2s^{\ast}}a^{\dagger
2}\right)\colon\exp\left\\{\left(\frac{1}{s^{\ast}}-1\right)a^{\dagger}a\right\\}\colon\exp\left(\frac{r^{\ast}}{2s^{\ast}}a^{2}\right)$
$\displaystyle=\exp\left(-\frac{r}{2s^{\ast}}a^{\dagger
2}\right)\exp\left\\{\left(a^{\dagger}a+\frac{1}{2}\right)\ln\frac{1}{s^{\ast}}\right\\}\exp\left(\frac{r^{\ast}}{2s^{\ast}}a^{2}\right),$
(97)

where we have used the mathematical formula [51]

$\displaystyle\int\frac{d^{2}z}{\pi}\exp\\{\zeta\left|z\right|^{2}+\xi z+\eta
z^{\ast}+fz^{2}+gz^{\ast 2}\\}$
$\displaystyle=\frac{1}{\sqrt{\zeta^{2}-4fg}}\exp\left\\{\frac{-\zeta\xi\eta+\xi^{2}g+\eta^{2}f}{\zeta^{2}-4fg}\right\\},$
(98)

with the convergent condition $\operatorname{Re}(\zeta\pm f\pm
g)<0,\operatorname{Re}(\frac{\zeta^{2}-4fg}{\zeta\pm f\pm g})<0.$ It then
follows

$\left\langle
z\right|F_{1}\left(s,r\right)\left|z^{\prime}\right\rangle=\frac{1}{\sqrt{s^{\ast}}}\exp\left[-\frac{\left|z\right|^{2}+\left|z^{\prime}\right|^{2}}{2}-\frac{rz^{\ast
2}}{2s^{\ast}}+\frac{r^{\ast}z^{\prime
2}}{2s^{\ast}}+\frac{z^{\ast}z^{\prime}}{s^{\ast}}\right].$ (99)

Then using

$\left\langle
x_{i}\right|\left.z\right\rangle=\pi^{-1/4}\exp\left(-\frac{x_{i}^{2}}{2}+\sqrt{2}x_{i}z-\frac{z^{2}}{2}-\frac{\left|z\right|^{2}}{2}\right).$
(100)

and the completeness relation of coherent state as well as (85) we obtain the
matrix element of $F_{1}\left(s,r\right)$ ($\equiv F_{1}\left(A,B,C\right)$)
in coordinate representation $\left\langle x_{i}\right|$,

$\displaystyle\left\langle
x_{2}\right|F_{1}\left(s,r\right)\left|x_{1}\right\rangle$
$\displaystyle=\int\frac{d^{2}z}{\pi}\left\langle
x_{2}\right|\left.z\right\rangle\left\langle
z\right|F_{1}\left(s,r\right)\int\frac{d^{2}z^{\prime}}{\pi}\left|z^{\prime}\right\rangle\left\langle
z^{\prime}\right|\left.x_{1}\right\rangle$ $\displaystyle=\frac{1}{\sqrt{2\pi
iB}}\exp\left[\frac{i}{2B}\left(Ax_{1}^{2}-2x_{2}x_{1}+Dx_{2}^{2}\right)\right]\equiv\mathcal{K}\left(x_{2},x_{1}\right),$
(101)

which is just the kernel of generalized Fresnel transform
$\mathcal{K}\left(x_{2},x_{1}\right)$ in (12). The above discussions
demonstrate how to transit classical Fresnel transform to GFO through the
coherent state and the IWOP technique.

Now if we define $g\left(x_{2}\right)=\left\langle
x_{2}\right|\left.g\right\rangle$, $f\left(x_{1}\right)=\left\langle
x_{1}\right|\left.f\right\rangle$ and using Eq. (17), we can rewrite Eq. (11)
as

$\displaystyle\left\langle x_{2}\right|\left.g\right\rangle$
$\displaystyle=\int_{-\infty}^{\infty}dx_{1}\left\langle
x_{2}\right|F_{1}\left(A,B,C\right)\left|x_{1}\right\rangle\left\langle
x_{1}\right|\left.f\right\rangle$ $\displaystyle=\left\langle
x_{2}\right|F_{1}\left(A,B,C\right)\left|f\right\rangle,$ (102)

which is just the quantum mechanical version of GFO. Therefore, the
1-dimensional GFT in classical optics corresponds to the 1-mode GFO
$F_{1}\left(A,B,C\right)$ operating on state vector $\left|f\right\rangle$ in
Hilbert space, i.e.
$\left|g\right\rangle=F_{1}\left(A,B,C\right)\left|f\right\rangle$. One merit
of GFO is: using coordinate-momentum representation transform we can
immediately obtain GFT in “frequency” domain, i.e.

$\displaystyle\left\langle
p_{2}\right|F\left|p_{1}\right\rangle=\int_{-\infty}^{\infty}dx_{1}dx_{2}\left\langle
p_{2}\right|\left.x_{2}\right\rangle\left\langle
x_{2}\right|F\left|x_{1}\right\rangle\left\langle
x_{1}\right.\left|p_{1}\right\rangle$ $\displaystyle=\frac{1}{\sqrt{2\pi
iB}}\int_{-\infty}^{\infty}\frac{dx_{1}dx_{2}}{2\pi}\exp\left[\frac{iA}{2B}\left(x_{1}^{2}+\frac{x_{1}}{A}\left(Bp_{1}-2x_{2}\right)\right)+\frac{iD}{2B}x_{2}^{2}-ip_{2}x_{2}\right]$
$\displaystyle=\frac{1}{\sqrt{2\pi
i\left(-C\right)}}\exp\left[\frac{i}{2\left(-C\right)}\left(Dp_{1}^{2}-2p_{2}p_{1}+Ap_{2}^{2}\right)\right].$
(103)

Obviously, $F_{1}\left(A,B,C\right)$ induces the following transform

$F_{1}^{-1}\left(A,B,C\right)\left(\begin{array}[c]{c}Q\\\
P\end{array}\right)F_{1}\left(A,B,C\right)=\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)\left(\begin{array}[c]{c}Q\\\ P\end{array}\right).$ (104)

###  5.2 Group Multiplication Rule for Single-mode GFO

Because two successive optical Fresnel transforms is still a Fresnel
transform, we wonder if the product of two GFO is still a GFO. On the other
hand, we have known that the GFO is the image of the symplectic transform
$z\rightarrow sz-rz^{\ast},$ we expect that the product of two symplectic
transforms maps into the GFO which is just the product of two GFOs. If this is
so, then correspondence between GFT and GFO is perfect. Using (96),
$\left\langle
z\right.\left|z^{\prime}\right\rangle=\exp\left[-\frac{1}{2}\left(|z|^{2}+|z^{\prime}|^{2}\right)+z^{\ast}z^{\prime}\right]$
and the IWOP technique we can directly perform the following integrals

$\displaystyle F_{1}\left(s,r\right)F_{1}\left(s^{\prime},r^{\prime}\right)$
$\displaystyle=\sqrt{ss^{\prime}}\int\frac{d^{2}zd^{2}z^{\prime}}{\pi^{2}}\left|sz-
rz^{\ast}\right\rangle\left\langle
z\right.\left|s^{\prime}z^{\prime}-r^{\prime}z^{\prime\ast}\right\rangle\left\langle
z^{\prime}\right|$
$\displaystyle=\frac{1}{\sqrt{s^{\prime\prime\ast}}}\exp\left[-\frac{r^{\prime\prime}}{2s^{\prime\prime\ast}}a^{\dagger
2}\right]\colon\exp\left\\{\left(\frac{1}{s^{\prime\prime\ast}}-1\right)a^{\dagger}a\right\\}\colon\exp\left[\frac{r^{\prime\prime\ast}}{2s^{\prime\prime\ast}}a^{2}\right]$
$\displaystyle=\sqrt{s^{\prime\prime}}\int\frac{d^{2}z}{\pi}\left|s^{\prime\prime\ast}z-r^{\prime\prime}z^{\ast}\right\rangle\left\langle
z\right|=F_{1}\left(s^{\prime\prime},r^{\prime\prime}\right),$ (105)

where we have set

$s^{\prime\prime}=ss^{\prime}+rr^{\prime\ast},\;r^{\prime\prime}=r^{\prime}s+rs^{\prime\ast},$
(106)

or

$M^{\prime\prime}\equiv\left(\begin{array}[c]{cc}s^{\prime\prime}&-r^{\prime\prime}\\\
-r^{\ast\prime\prime}&s^{\ast\prime\prime}\end{array}\right)=\left(\begin{array}[c]{cc}s&-r\\\
-r^{\ast}&s^{\ast}\end{array}\right)\left(\begin{array}[c]{cc}s^{\prime}&-r^{\prime}\\\
-r^{\prime\ast}&s^{\prime\ast}\end{array}\right)=MM^{\prime},\
\left|s^{\prime\prime}\right|^{2}-\left|r^{\prime\prime}\right|^{2}=1,$ (107)

from which we see that it is just the mapping of the above ($A,B,C,D)$
matrices multiplication. Hence
$F_{1}\left(s,r\right)F_{1}\left(s^{\prime},r^{\prime}\right)$ is the loyal
representation of the product of two symplectic group elements shown in (107).
The above discussion actually reveals an important property of coherent
states, though two coherent state vectors are not orthogonal, but the equation

$\sqrt{ss^{\prime}}\int\frac{d^{2}zd^{2}z^{\prime}}{\pi^{2}}\left|sz-
rz^{\ast}\right\rangle\left\langle
z\right.\left|s^{\prime}z^{\prime}-r^{\prime}z^{\prime\ast}\right\rangle\left\langle
z^{\prime}\right|=\sqrt{s^{\prime\prime}}\int\frac{d^{2}z}{\pi}\left|s^{\prime\prime\ast}z-r^{\prime\prime}z^{\ast}\right\rangle\left\langle
z\right|$ (108)

seems as if their overlap $\left\langle
z\right.\left|s^{\prime}z^{\prime}-r^{\prime}z^{\prime\ast}\right\rangle$ was
a $\delta$-function. The coherent state representation for GFOs’ product may
be visualized very easily, but it achieves striking importance, because it
does not change its form when treating symplectic transform according to
$z\rightarrow sz-rz^{\ast}$.

As a result of this group multiplication rule of GFO, we immediately obtain

$\displaystyle\mathcal{K}^{M^{\prime\prime}}\left(x_{2},x_{1}\right)=\left\langle
x_{2}\right|F_{1}\left(s^{\prime\prime},r^{\prime\prime}\right)\left|x_{1}\right\rangle$
$\displaystyle=\int_{-\infty}^{\infty}dx_{3}\left\langle
x_{2}\right|F_{1}\left(s,r\right)\left|x_{3}\right\rangle\left\langle
x_{3}\right|F_{1}\left(s^{\prime},r^{\prime}\right)\left|x_{1}\right\rangle$
$\displaystyle=\int_{-\infty}^{\infty}dx_{3}\mathcal{K}^{M}\left(x_{2},x_{3}\right)\mathcal{K}^{M^{\prime}}\left(x_{3},x_{1}\right),$
(109)

provided that the parameter matrices
$\left(s^{\prime\prime},r^{\prime\prime}\right)$ satisfy (106). Thus by virtue
of the group multiplication property of GFO we immediately find the successive
transform property of GFTs.

##  6 Quantum Optical ABCD Law for optical propagation —single-mode case

In classical optics, ray-transfer matrices,
$N=\left(\begin{array}[c]{cc}A&B\\\ C&D\end{array}\right),$ $AD-BC=1$, have
been used to describe the geometrical formation of images by a centered lens
system. For an optical ray (a centered spherical wavefront) passing through
optical instruments there is a famous law, named ABCD law, governing the
relation between input ray $\left(r_{1},\alpha_{1}\right)$ and output ray
$\left(r_{2},\alpha_{2}\right),$ i.e.

$\left(\begin{array}[c]{c}r_{2}\\\
\alpha_{2}\end{array}\right)=N\left(\begin{array}[c]{c}r_{1}\\\
\alpha_{1}\end{array}\right),$ (110)

where $r_{1}$ is the ray height from the optical axis, and $\alpha_{1}$ is
named the optical direction-cosine, $r_{1}/\alpha_{1}\equiv R_{1}$ specifies
the ray’s wavefront shape. Eq. (110) implies

$R_{2}\equiv\frac{r_{2}}{\alpha_{2}}=\frac{AR_{1}+B}{CR_{1}+D}.$ (111)

This law is the core of matrix optics, since it tells us how the curvature of
a centered spherical wavefront changes from one reference plane to the next.
Besides, the multiplication rule of matrix optics implies that if the ray-
transfer matrices of the $n$ optical components are
$N_{1},N_{2},N_{3},\cdots,N_{n}$, respectively, then the whole system is
determined by a matrix $N=N_{1}N_{2}N_{3}\cdots N_{n}.$

One of the remarkable things of modern optics is the case with which
geometrical ray-transfer methods, constituting the matrix optics, can be
adapted to describe the generation and propagation of Laser beams. In 1965
Kogelnik [52] pointed out that propagation of Gaussian beam also obeys ABCD
law via optical diffraction integration, i.e. the input light field
$f\left(x_{1}\right)$ and output light field $g\left(x_{2}\right)$ are related
to each other by so-called Fresnel integration [19]
$g\left(x_{2}\right)=\int_{-\infty}^{\infty}\mathcal{K}\left(A,B,C;x_{2},x_{1}\right)f\left(x_{1}\right)dx_{1},$
where

$\mathcal{K}\left(A,B,C;x_{2},x_{1}\right)=\frac{1}{\sqrt{2\pi
iB}}\exp\left[\frac{i}{2B}\left(Ax_{1}^{2}-2x_{2}x_{1}+Dx_{2}^{2}\right)\right].$

The ABCD law for Gaussian beam passing through an optical system is [53]

$q_{2}=\frac{Aq_{1}+B}{Cq_{1}+D},$ (112)

where $q_{1}$ $(q_{2})$ represents the complex curvature of the input (output)
Gaussian beam, Eq. (112) has the similar form as Eq. (111). An interesting and
important question naturally arises [54]: Does ABCD law also exhibit in
quantum optics? Since classical Fresnel transform should have its quantum
optical counterpart?

To see the ABCD law more explicitly, using Eq.(85) we can re-express Eq.(97)
as

$\displaystyle F_{1}\left(A,B,C\right)$
$\displaystyle=\sqrt{\frac{2}{A+D+i\left(B-C\right)}}\colon\exp\left\\{\frac{A-D+i\left(B+C\right)}{2\left[A+D+i\left(B-C\right)\right]}a^{\dagger
2}\right.$
$\displaystyle\left.+\left[\frac{2}{A+D+i\left(B-C\right)}-1\right]a^{\dagger}a-\frac{A-D-i\left(B+C\right)}{2\left[A+D+i\left(B-C\right)\right]}a^{2}\right\\}\colon,$
(113)

and the multiplication rule for $F_{1}$ is
$F\left(A^{\prime},B^{\prime},C^{\prime},D^{\prime}\right)F\left(A,B,C,D\right)=F\left(A^{\prime\prime},B^{\prime\prime},C^{\prime\prime},D^{\prime\prime}\right),$
where

$\left(\begin{array}[c]{cc}A^{\prime\prime}&B^{\prime\prime}\\\
C^{\prime\prime}&D^{\prime\prime}\end{array}\right)=\left(\begin{array}[c]{cc}A^{\prime}&B^{\prime}\\\
C^{\prime}&D^{\prime}\end{array}\right)\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right).$ (114)

Next we directly use the GFO to derive ABCD law in quantum optics. From
Eq.(114) we see that the GFO generates

$F_{1}\left(A,B,C\right)\left|0\right\rangle=\sqrt{\frac{2}{A+iB-i\left(C+iD\right)}}\exp\left\\{\frac{A-D+i\left(B+C\right)}{2\left[A+D+i\left(B-C\right)\right]}a^{\dagger
2}\right\\}\left|0\right\rangle,$ (115)

if we identify

$\frac{A-D+i\left(B+C\right)}{A+D+i\left(B-C\right)}=\frac{q_{1}-i}{q_{1}+i},$
(116)

then

$F_{1}\left(A,B,C\right)\left|0\right\rangle=\sqrt{-\frac{2/\left(C+iD\right)}{q_{1}+i}}\exp\left[\frac{q_{1}-i}{2\left(q_{1}+i\right)}a^{\dagger
2}\right]\left|0\right\rangle,$ (117)

The solution of Eq.(116) is

$q_{1}\equiv-\frac{A+iB}{C+iD}.$ (118)

Let $F_{1}\left(A,B,C\right)\left|0\right\rangle$ expressed by (117) be an
input state for an optical system which is characteristic by parameters
$A^{\prime},B^{\prime},C^{\prime},D^{\prime},$ then the quantum optical ABCD
law states that the output state is

$F_{1}\left(A^{\prime},B^{\prime},C^{\prime}\right)F_{1}\left(A,B,C\right)\left|0\right\rangle=\sqrt{\frac{-2/\left(C^{\prime\prime}+iD^{\prime\prime}\right)}{q_{2}+i}}\exp\left[\frac{q_{2}-i}{2\left(q_{2}+i\right)}a^{\dagger
2}\right]\left|0\right\rangle,$ (119)

which has the similar form as Eq.(117), where
$\left(C^{\prime\prime},D^{\prime\prime}\right)$ is determined by Eq.(114),
and

$\bar{q}_{2}=\frac{A^{\prime}\bar{q}_{1}+B^{\prime}}{C^{\prime}\bar{q}_{1}+D^{\prime}},\text{
\ }\bar{q}_{j}\equiv-q_{j},\text{ \ }\left(j=1,2\right)$ (120)

which resembles Eq.(112).

Proof:

According to the multiplication rule of two GFOs and Eqs.(113)-(114) we have

$\displaystyle
F_{1}\left(A^{\prime},B^{\prime},C^{\prime}\right)F_{1}\left(A,B,C\right)\left|0\right\rangle$
$\displaystyle=\sqrt{\frac{2}{A^{\prime\prime}+D^{\prime\prime}+i\left(B^{\prime\prime}-C^{\prime\prime}\right)}}\exp\left\\{\frac{A^{\prime\prime}-D^{\prime\prime}+i\left(B^{\prime\prime}+C^{\prime\prime}\right)}{2\left[A^{\prime\prime}+D^{\prime\prime}+i\left(B^{\prime\prime}-C^{\prime\prime}\right)\right]}a^{\dagger
2}\right\\}\left|0\right\rangle$
$\displaystyle=\sqrt{\frac{2}{A^{\prime}\left(A+iB\right)+B^{\prime}\left(C+iD\right)-iC^{\prime}\left(A+iB\right)-iD^{\prime}\left(C+iD\right)}}$
$\displaystyle\times\exp\left\\{\frac{A^{\prime}\left(A+iB\right)+B^{\prime}\left(C+iD\right)+iC^{\prime}\left(A+iB\right)+iD^{\prime}\left(C+iD\right)}{2\left[A^{\prime}\left(A+iB\right)+B^{\prime}\left(C+iD\right)-iC^{\prime}\left(A+iB\right)-iD^{\prime}\left(C+iD\right)\right]}a^{\dagger
2}\right\\}\left|0\right\rangle$
$\displaystyle=\sqrt{\frac{-2/\left(C+iD\right)}{A^{\prime}q_{1}-B^{\prime}-i\left(C^{\prime}q_{1}-D^{\prime}\right)}}\exp\left\\{\frac{A^{\prime}q_{1}-B^{\prime}+i\left(C^{\prime}q_{1}-D^{\prime}\right)}{2\left[A^{\prime}q_{1}-B^{\prime}-i\left(C^{\prime}q_{1}-D^{\prime}\right)\right]}a^{\dagger
2}\right\\}\left|0\right\rangle.$ (121)

Using Eq.(118) we see
$\frac{2/\left(C+iD\right)}{C^{\prime}q_{1}-D^{\prime}}=-2/\left(C^{\prime\prime}+iD^{\prime\prime}\right),$
together using Eq.(120) we can reach Eq.(119), thus the law is proved. Using
Eq. (117) we can re-express Eq.(120) as

$q_{2}=-\frac{A^{\prime}(A+iB)+B^{\prime}(C+iD)}{C^{\prime}(A+iB)+D^{\prime}(C+iD)}=-\frac{A^{\prime\prime}+iB^{\prime\prime}}{C^{\prime\prime}+iD^{\prime\prime}},$
(122)

which is in consistent to Eq.(118). Eqs. (117)-(122) are therefore self-
consistent.

As an application of quantum optical ABCD law, we apply it to tackle the time-
evolution of a time-dependent harmonic oscillator whose Hamiltonian is

$H=\frac{1}{2}e^{-2\gamma t}P^{2}+\frac{1}{2}\omega_{0}^{2}e^{2\gamma
t}Q^{2},\text{ \ \ }\hbar=1,$ (123)

where we have set the initial mass $m_{0}=1,$ $\gamma$ denotes damping. Using
$u\left(t\right)=e^{\frac{i\gamma}{2}Q^{2}}e^{-\frac{i\gamma
t}{2}\left(QP+PQ\right)}\ $to perform the transformation

$\displaystyle u\left(t\right)Qu^{-1}\left(t\right)$ $\displaystyle=e^{-\gamma
t}Q,$ $\displaystyle u\left(t\right)Pu^{-1}\left(t\right)$
$\displaystyle=e^{\gamma t}P-\gamma e^{\gamma t}Q,$ (124)

then $i\frac{\partial\left|\psi\left(t\right)\right\rangle}{\partial
t}=H\left|\psi\left(t\right)\right\rangle\ $leads to
$i\frac{\partial\left|\phi\right\rangle}{\partial
t}=\mathcal{H}\left|\phi\right\rangle,$
$\left|\phi\right\rangle=u\left(t\right)\left|\psi\left(t\right)\right\rangle,$

$H\rightarrow\mathcal{H}=u\left(t\right)Hu^{-1}\left(t\right)-iu\left(t\right)\frac{\partial
u^{-1}\left(t\right)}{\partial
t}=\frac{1}{2}P^{2}+\frac{1}{2}\omega^{2}Q^{2},.$ (125)

where $\omega^{2}=\omega_{0}^{2}-\gamma^{2}.$ $\mathcal{H}$ does not contain
$t$ explicitly. The dynamic evolution of a mass-varying harmonic oscillator
from the Fock state $\left|0\right\rangle$ at initial time to a squeezed state
at time $t$ is

$\left|\psi\left(t\right)\right\rangle_{0}=u^{-1}\left(t\right)\left|0\right\rangle=e^{\frac{i\gamma
t}{2}\left(QP+PQ\right)}e^{-\frac{i\gamma}{2}Q^{2}}\left|0\right\rangle,$
(126)

if we let $A=D=1,B=0,C=-\gamma;$ and $A^{\prime}=e^{-\gamma
t},D^{\prime}=e^{\gamma t},B^{\prime}=C^{\prime}=0,$ then
$q_{1}=\frac{1}{\gamma-i},$ $q_{2}=\frac{e^{-2\gamma t}}{\gamma-i}$, according
to Eq.(119) we directly obtain

$u^{-1}\left(t\right)\left|0\right\rangle=\sqrt{\frac{2e^{-\gamma
t}}{e^{-2\gamma t}+i\gamma+1}}\exp\left[\frac{e^{-2\gamma
t}-1-i\gamma}{2\left(e^{-2\gamma t}+1+i\gamma\right)}a^{\dagger
2}\right]\left|0\right\rangle,$ (127)

so the time evolution of the damping oscillator embodies the quantum optical
ABCD law.

##  7 Optical operator method studied via GFO’s decomposition

Fresnel diffraction is the core of Fourier optics [19, 30, 35, 36], Fresnel
transform is frequently used in optical imaging, optical propagation and
optical instrument design. The GFT represents a class of optical transforms
which are of great importance for their applications to describe various
optical systems. It is easily seen that when we let the transform kernel
$\mathcal{K}\left(x_{2},x_{1}\right)=\exp\left(ix_{2}x_{1}\right)$, the GFT
changes into the well-known Fourier transform, which is adapted to express
mathematically the Fraunhofer diffraction. And if
$\mathcal{K}\left(x_{2},x_{1}\right)=\exp[i\left(x_{2}-x_{1}\right)^{2}]$, the
GFT then describes a Fresnel diffraction. In studying various optical
transformations one also proposed so-called optical operator method [55] which
used quantum mechanical operators’ ordered product to express the mechanism of
optical systems, such that the ray transfer through optical instruments and
the diffraction can be discussed by virtue of the commutative relations of
operators and the matrix algebra. Two important questions thus naturally
arises: how to directly map the classical optical transformations to the
optical operator method? How to combine the usual optical transformation
operators, such as the square phase operators, scaling operator, Fourier
transform operator and the propagation operator in free space, into a concise
and unified form? In this section we shall solve these two problems and
develop the optical operator method onto a new stage.

###  7.1 Four fundamental optical operators derived by decomposing GFO

The GFO $F_{1}\left(A,B,C\right)$ can also be expressed in the form of
quadratic combination of canonical operators $Q$ and $P$ [56], i.e.,

$F_{1}\left(A,B,C\right)=\exp\left(\frac{iC}{2A}Q^{2}\right)\exp\left(-\frac{i}{2}\left(QP+PQ\right)\ln
A\right)\exp\left(-\frac{iB}{2A}P^{2}\right),$ (128)

where we have set $\hbar=1,$ $A\neq 0.$ To confirm this, we first calculate
matrix element

$\displaystyle\left\langle
x\right|F_{1}\left(A,B,C\right)\left|p\right\rangle$
$\displaystyle=\exp\left(\frac{iC}{2A}x^{2}\right)\left\langle
x\right|\exp\left(-\frac{i}{2}\left(QP+PQ\right)\ln
A\right)\left|p\right\rangle\exp\left(-\frac{iB}{2A}p^{2}\right)$
$\displaystyle=\frac{1}{\sqrt{2\pi
A}}\exp\left(\frac{iC}{2A}x^{2}-\frac{iB}{2A}p^{2}+\frac{ipx}{A}\right),$
(129)

where we have used the squeezing property

$\exp\left[-\frac{i}{2}\left(QP+PQ\right)\ln
A\right]\left|p\right\rangle=\frac{1}{\sqrt{A}}\left|p/A\right\rangle.$ (130)

It then follows from (129) and $AD-BC=1$, we have

$\left\langle
x_{2}\right|F_{1}\left(A,B,C\right)\left|x_{1}\right\rangle=\int_{-\infty}^{\infty}dp\left\langle
x_{2}\right|F_{1}\left(A,B,C\right)\left|p\right\rangle\left\langle
p\right|\left.x_{1}\right\rangle=\mathcal{K}^{M}\left(x_{2},x_{1}\right).$
(131)

Thus $F_{1}\left(A,B,C\right)$ in (128) is really the expected GFO. Next we
directly use (84) and the canonical operator $\left(Q,P\right)$ representation
(128) to develop the optical operator method.

By noticing the matrix decompositions [31]

$\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)=\left(\begin{array}[c]{cc}1&0\\\
C/A&1\end{array}\right)\left(\begin{array}[c]{cc}A&0\\\
0&A^{-1}\end{array}\right)\left(\begin{array}[c]{cc}1&B/A\\\
0&1\end{array}\right),$ (132)

and comparing (84) and (128) as well as using (114) we know

$F_{1}\left(A,B,C\right)=F_{1}\left(1,0,C/A\right)F_{1}\left(A,0,0\right)F_{1}\left(1,B/A,0\right),$
(133)

where

$\displaystyle F_{1}\left(1,0,C/A\right)$
$\displaystyle=\frac{\sqrt{2+iC/A}}{2\sqrt{2}\pi}\int
dxdp\left|\left(\begin{array}[c]{cc}1&0\\\
C/A&1\end{array}\right)\left(\begin{array}[c]{c}x\\\
p\end{array}\right)\right\rangle\left\langle\left(\begin{array}[c]{c}x\\\
p\end{array}\right)\right|$ (140)
$\displaystyle=\exp\left(\frac{iC}{2A}Q^{2}\right),$ (141)

which is named quadrature phase operator; and

$\displaystyle F_{1}\left(1,B/A,0\right)$
$\displaystyle=\frac{\sqrt{2-iB/A}}{2\sqrt{2}\pi}\int
dxdp\left|\left(\begin{array}[c]{cc}1&B/A\\\
0&1\end{array}\right)\left(\begin{array}[c]{c}x\\\
p\end{array}\right)\right\rangle\left\langle\left(\begin{array}[c]{c}x\\\
p\end{array}\right)\right|$ (148)
$\displaystyle=\exp\left(-\frac{iB}{2A}P^{2}\right),$ (149)

which is named Fresnel propagator in free space; as well as

$\displaystyle F_{1}\left(A,0,0\right)$
$\displaystyle=\frac{\sqrt{A+A^{-1}}}{2\sqrt{2}\pi}\int
dxdp\left|\left(\begin{array}[c]{cc}A&0\\\
0&A^{-1}\end{array}\right)\left(\begin{array}[c]{c}x\\\
p\end{array}\right)\right\rangle\left\langle\left(\begin{array}[c]{c}x\\\
p\end{array}\right)\right|$ (156)
$\displaystyle=\exp\left[-\frac{i}{2}\left(QP+PQ\right)\ln A\right],$ (157)

which is named scaling operator (squeezed operator [9, 10]). When
$A=D=0,B=1,C=-1,$ from (113) we see

$\displaystyle F_{1}\left(0,1,-1\right)$
$\displaystyle=\sqrt{-i}\int\frac{dxdp}{2\pi}\left|\left(\begin{array}[c]{cc}0&1\\\
-1&0\end{array}\right)\left(\begin{array}[c]{c}x\\\
p\end{array}\right)\right\rangle\left\langle\left(\begin{array}[c]{c}x\\\
p\end{array}\right)\right|$ (164)
$\displaystyle=\exp\left[-\left(a^{\dagger}a+\frac{1}{2}\right)\ln i\right]$
$\displaystyle=\exp\left[-i\frac{\pi}{2}\left(a^{\dagger}a+\frac{1}{2}\right)\right],$
(165)

which is named the Fourier operator, since it quantum mechanically transforms
[57]

$\displaystyle\exp\left[i\frac{\pi}{2}\left(a^{\dagger}a+\frac{1}{2}\right)\right]Q\exp\left[-i\frac{\pi}{2}\left(a^{\dagger}a+\frac{1}{2}\right)\right]$
$\displaystyle=P,$
$\displaystyle\exp\left[i\frac{\pi}{2}\left(a^{\dagger}a+\frac{1}{2}\right)\right]P\exp\left[-i\frac{\pi}{2}\left(a^{\dagger}a+\frac{1}{2}\right)\right]$
$\displaystyle=-Q.$ (166)

###  7.2 Alternate decompositions of GFO

Note that when $A=0$, the decomposition (128) is not available, instead, from

$\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)^{-1}=\left(\begin{array}[c]{cc}D&-B\\\
-C&A\end{array}\right),$ (167)

and (113), (114) and (128) we have

$F_{1}^{-1}\left(A,B,C\right)=\exp\left(-\frac{iC}{2D}Q^{2}\right)\exp\left[-\frac{i}{2}\left(QP+PQ\right)\ln
D\right]\exp\left(\frac{iB}{2D}P^{2}\right),$ (168)

it then follows

$F_{1}\left(A,B,C\right)=\exp\left(-\frac{iB}{2D}P^{2}\right)\exp\left[\frac{i}{2}\left(QP+PQ\right)\ln
D\right]\exp\left(\frac{iC}{2D}Q^{2}\right),\text{ }D\neq 0.$ (169)

Besides, when we notice

$\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)=\left(\begin{array}[c]{cc}1&0\\\
D/B&1\end{array}\right)\left(\begin{array}[c]{cc}B&0\\\
0&1/B\end{array}\right)\left(\begin{array}[c]{cc}0&1\\\
-1&0\end{array}\right)\left(\begin{array}[c]{cc}1&0\\\
A/B&1\end{array}\right),$ (170)

and

$\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)=\left(\begin{array}[c]{cc}1&A/C\\\
0&1\end{array}\right)\left(\begin{array}[c]{cc}-1/C&0\\\
0&-C\end{array}\right)\left(\begin{array}[c]{cc}0&1\\\
-1&0\end{array}\right)\left(\begin{array}[c]{cc}1&D/C\\\
0&1\end{array}\right),$ (171)

we have another decomposition for $B\neq 0,$

$\displaystyle F_{1}\left(A,B,C\right)$
$\displaystyle=\exp\left(\frac{iD}{2B}Q^{2}\right)\exp\left(-\frac{i}{2}\left(QP+PQ\right)\ln
B\right)$
$\displaystyle\times\exp\left[-\frac{i\pi}{2}\left(a^{\dagger}a+\frac{1}{2}\right)\right]\exp\left(\frac{iA}{2B}Q^{2}\right),\text{
}$ (172)

and for $C\neq 0$

$\displaystyle F_{1}\left(A,B,C\right)$
$\displaystyle=\exp\left(-\frac{iA}{2C}P^{2}\right)\exp\left[-\frac{i}{2}\left(QP+PQ\right)\ln\left(\frac{-1}{C}\right)\right]$
$\displaystyle\times\exp\left[-\frac{i\pi}{2}\left(a^{\dagger}a+\frac{1}{2}\right)\right]\exp\left(-\frac{iD}{2C}P^{2}\right).$
(173)

###  7.3 Some optical operator identities

For a special optical systems with the parameter $A=0,$ $C=-B^{-1},$

$\left(\begin{array}[c]{cc}0&B\\\
-B^{-1}&D\end{array}\right)=\left(\begin{array}[c]{cc}1&0\\\
D/B&1\end{array}\right)\left(\begin{array}[c]{cc}B&0\\\
0&B^{-1}\end{array}\right)\left(\begin{array}[c]{cc}0&1\\\
-1&0\end{array}\right),$ (174)

we have

$\displaystyle\exp\left(-\frac{iB}{2D}P^{2}\right)\exp\left(\frac{i}{2}\left(QP+PQ\right)\ln
D\right)\exp\left(\frac{-i}{2DB}Q^{2}\right)$
$\displaystyle=\exp\left(\frac{-iD}{2B}Q^{2}\right)\exp\left(\frac{i}{2}\left(QP+PQ\right)\ln
B\right)\exp\left[-\frac{i\pi}{2}\left(a^{\dagger}a+\frac{1}{2}\right)\right].$
(175)

In particular, when $A=D=0,$ $C=-B^{-1},$ from

$\left(\begin{array}[c]{cc}0&B\\\
-\frac{1}{B}&0\end{array}\right)=\left(\begin{array}[c]{cc}B&0\\\
0&\frac{1}{B}\end{array}\right)\left(\begin{array}[c]{cc}0&1\\\
-1&0\end{array}\right),$ (176)

we have

$\displaystyle\exp\left[-\frac{B^{2}-1}{2\left(B^{2}+1\right)}a^{\dagger
2}\right]\exp\left[\left(a^{\dagger}a+\frac{1}{2}\right)\ln\left(\frac{-2Bi}{B^{2}+1}\right)\right]\exp\left[-\frac{B^{2}-1}{2\left(B^{2}+1\right)}a^{2}\right]$
$\displaystyle=\exp\left[-\frac{i}{2}\left(QP+PQ\right)\ln
B\right]\exp\left[-i\frac{\pi}{2}\left(a^{\dagger}a+\frac{1}{2}\right)\right].$
(177)

Using the following relations

$\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)=\left(\begin{array}[c]{cc}1&\left(A-1\right)/C\\\
0&1\end{array}\right)\left(\begin{array}[c]{cc}1&0\\\
C&1\end{array}\right)\left(\begin{array}[c]{cc}1&\left(D-1\right)/C\\\
0&1\end{array}\right),$ (178)

it then follows that

$F_{1}\left(A,B,C\right)=\exp\left(-\frac{i\left(A-1\right)}{2C}P^{2}\right)\exp\left(\frac{iC}{2}Q^{2}\right)\exp\left(-\frac{i\left(D-1\right)}{2C}P^{2}\right),\text{
}$ (179)

while from

$\left(\begin{array}[c]{cc}A&B\\\
C&D\end{array}\right)=\left(\begin{array}[c]{cc}1&0\\\
\left(D-1\right)/B&1\end{array}\right)\left(\begin{array}[c]{cc}1&B\\\
0&1\end{array}\right)\left(\begin{array}[c]{cc}1&0\\\
\left(A-1\right)/B&1\end{array}\right)$ (180)

we obtain

$F_{1}\left(A,B,C\right)=\exp\left(\frac{i\left(D-1\right)}{2B}Q^{2}\right)\exp\left(-\frac{iB}{2}P^{2}\right)\exp\left(\frac{i\left(A-1\right)}{2B}Q^{2}\right),\text{
}$ (181)

so we have

$\displaystyle\exp\left(\frac{i\left(D-1\right)}{2B}Q^{2}\right)\exp\left(-\frac{iB}{2}P^{2}\right)\exp\left(\frac{i\left(A-1\right)}{2B}Q^{2}\right)$
$\displaystyle=\exp\left(-\frac{i\left(A-1\right)}{2C}P^{2}\right)\exp\left(\frac{iC}{2}Q^{2}\right)\exp\left(-\frac{i\left(D-1\right)}{2C}P^{2}\right).$
(182)

In this section, based on a one-to-one correspondence between classical
Fresnel transform in phase space and quantum unitary transform in state-vector
space and the IWOP technique as well as the coherent state representation we
have found a way to directly map the classical optical transformations to the
optical operator method. We have combined the usual optical transformation
operators, such as the square phase operators, scaling operator, Fourier
transform operator and the propagation operator in free space, into a concise
and unified form. The various decompositions of Fresnel operator into the
exponential canonical operators are also obtained.

##  8 Quantum tomography and probability distribution for the Fresnel
quadrature phase

In quantum optics theory all possible linear combinations of quadratures $Q$
and $P$ of the oscillator field mode $a$ and $a^{\dagger}$ 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
oscillator phase, is connected with the marginal distribution of Wigner
function (WF), thus the homodyne measurement of light field permits the
reconstruction of the WF of a quantum system by varying the phase shift
between two oscillators. In Ref. [58] Vogel and Risken pointed out that the
probability distribution for the rotated quadrature phase
$Q_{\theta}\equiv[a^{\dagger}\exp(i\theta)+a\exp(-i\theta)]/\sqrt{2},$
$\left[a,a^{\dagger}\right]=1,$which depends on only one $\theta$ angle, can
be expressed in terms of WF, and that the reverse is also true (named as
Vogel-Risken relation), i.e., one can obtain the Wigner distribution by
tomographic inversion of a set of measured probability distributions,
$P_{\theta}\left(q_{\theta}\right),$ of the quadrature amplitude. Once the
distribution $P_{\theta}\left(q_{\theta}\right)$ are obtained, one can use the
inverse Radon transformation familiar in tomographic imaging to obtain the
Wigner distribution and density matrix. The Radon transform of the WF 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 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 slice image is equated with tomogram.) The theoretical development in
quantum tomography in the last decade has progressed in the direction of
determining more physical relevant parameters of the density from tomographic
data[58, 59, 60, 61, 62].

###  8.1 Relation between Fresnel transform and Radon transform of WF

In [63, 64] the Radon transform of WF which depends on two continuous
parameters is introduced, this has the advantage in conveniently associating
quantum tomography theory with squeezed coherent state theory. In this
subsection we want to derive relations between the Fresnel transform and the
Radon transform of WF in quantum optics in tomography theory.

By extending the rotated quadrature phase $Q_{\theta}$ to the Fresnel
quadrature phase

$Q_{F}\equiv\left(s^{\ast}a+ra^{\dagger}+sa^{\dagger}+r^{\ast}a\right)/\sqrt{2}=F_{1}QF_{1}^{\dagger},$
(183)

where $s$ and $r$ are related to ABCD through(85),

$s=\frac{1}{2}\left[A+D-i\left(B-C\right)\right],\;r=-\frac{1}{2}\left[A-D+i\left(B+C\right)\right],\;|s|^{2}-|r|^{2}=1,$
(184)

we shall prove that the $(D,B)$ related Radon transform of Wigner operator
$\Delta\left(q,p\right)$ is just the pure state density operator
$\left|q\right\rangle_{s,rs,r}\left\langle q\right|$ (named as the tomographic
density operator) formed with the eigenstates belonging to the quadrature
$Q_{F}$, ( $\left|q\right\rangle_{s,r}=F_{1}\left|q\right\rangle,$ $Q$ is the
coordinate operator),

$F_{1}\left|q\right\rangle\left\langle
q\right|F_{1}^{\dagger}=\left|q\right\rangle_{s,rs,r}\left\langle
q\right|=\int_{-\infty}^{\infty}dq^{\prime}dp^{\prime}\delta\left[q-\left(Dq^{\prime}-Bp^{\prime}\right)\right]\Delta\left(q^{\prime},p^{\prime}\right),$
(185) $D=\frac{1}{2}\left(s+s^{\ast}+r+r^{\ast}\right),\
B=\frac{1}{2i}\left(s^{\ast}-s+r^{\ast}-r\right),$ (186)

Since $F$ corresponds to classical Fresnel transform in optical diffraction
theory, so Eq. (185) indicates that the probability distribution for the
Fresnel quadrature phase is the Radon transform of WF [65].

Proof:

Firstly, from (97) we see

$F_{1}\left(s,r\right)aF_{1}^{{\dagger}}\left(s,r\right)=s^{\ast}a+ra^{\dagger},$
(187)

so from $Q=\frac{a+a^{\dagger}}{\sqrt{2}},P=i\frac{a^{\dagger}-a}{\sqrt{2}},$
indeed we have

$F_{1}QF_{1}^{\dagger}=F_{1}\frac{a+a^{\dagger}}{\sqrt{2}}F_{1}^{\dagger}=\left(s^{\ast}a+ra^{\dagger}+sa^{\dagger}+r^{\ast}a\right)/\sqrt{2}=Q_{F}.$
(188)

Secondly, we can derive the explicit form of $\left|q\right\rangle_{s,r}.$
Starting from $s^{\ast}+r^{\ast}=D+iB,$ $s^{\ast}-r^{\ast}=A-iC,$ we set up
the eigenvector equation

$Q_{F}\left|q\right\rangle_{s,r}=\left(DQ-
BP\right)\left|q\right\rangle_{s,r}=q\left|q\right\rangle_{s,r},\text{ }$
(189)

it follows

$\left|q\right\rangle_{s,r}=F_{1}\left(s,r\right)\left|q\right\rangle.$ (190)

In the coordinate and momentum representations we have

$\displaystyle\left\langle q^{\prime}\right|Q_{F}\left|q\right\rangle_{s,r}$
$\displaystyle=\left(Dq^{\prime}+iB\frac{d}{dq^{\prime}}\right)\left\langle
q^{\prime}\right|\left.q\right\rangle_{s,r}=q\left\langle
q^{\prime}\right|\left.q\right\rangle_{s,r}.$ (191) $\displaystyle\left\langle
p\right|Q_{F}\left|q\right\rangle_{s,r}$
$\displaystyle=\left(iD\frac{d}{dp}-Bp\right)\left\langle
p\right|\left.q\right\rangle_{s,r}=q\left\langle
p\right|\left.q\right\rangle_{s,r}.$ (192)

The normalizable solutions to (191) and (192) are

$\displaystyle\left\langle q^{\prime}\right|\left.q\right\rangle_{s,r}$
$\displaystyle=c\left(q\right)\exp\left[\frac{iq^{\prime}\left(Dq^{\prime}-2q\right)}{2B}\right],$
(193) $\displaystyle\left\langle p\right|\left.q\right\rangle_{s,r}$
$\displaystyle=d\left(q\right)\exp\left[\frac{ip\left(-Bp-2q\right)}{2D}\right].$
(194)

Using the Fock representation of $\left|q\right\rangle$ and
$\left|p\right\rangle$ in Eqs.(14) and (23), we obtain

$\displaystyle\left|q\right\rangle_{s,r}$
$\displaystyle=\int_{-\infty}^{\infty}dq^{\prime}\left|q^{\prime}\right\rangle\left\langle
x^{\prime}\right|\left.q\right\rangle_{s,r}$
$\displaystyle=\pi^{-1/4}c\left(q\right)\sqrt{\frac{2B\pi}{B-iD}}\exp\left[-\frac{q^{2}}{2B\left(B-iD\right)}+\frac{\sqrt{2}a^{\dagger}q}{D+iB}-\frac{D-iB}{D+iB}\frac{a^{\dagger
2}}{2}\right]\left|0\right\rangle,$ (195)

and

$\displaystyle\left|q\right\rangle_{s,r}$
$\displaystyle=\int_{-\infty}^{\infty}dp\left|p\right\rangle\left\langle
p\right|\left.q\right\rangle_{s,r}$
$\displaystyle=d\left(q\right)\pi^{-1/4}\sqrt{\frac{2\pi
D}{D+iB}}\exp\left[-\frac{q^{2}}{2D\left(D+iB\right)}+\allowbreak\frac{\sqrt{2}a^{\dagger}q}{D+iB}-\frac{D-iB}{D+iB}\frac{a^{\dagger
2}}{2}\right]\left|0\right\rangle.$ (196)

Comparing Eq.(195) with (196) we see

$\frac{c\left(q\right)}{d\left(q\right)}=\sqrt{\allowbreak\frac{D}{iB}}\exp\left[\frac{\allowbreak
iA}{2B}q^{2}-\frac{iCq^{2}}{2D}\right].$ (197)

On the other hand, according to the orthogonalization of
$\left|q\right\rangle_{s,r}$, ${}_{s,r}\left\langle
q^{\prime}\right.\left|q^{\prime\prime}\right\rangle_{s,r}=\delta\left(q^{\prime}-q^{\prime\prime}\right),$
we have

$\left|c\left(q\right)\right|^{2}=\frac{1}{2\pi B},\text{ \
}\left|d\left(q\right)\right|^{2}=\frac{1}{2\pi D}.$ (198)

Thus combining Eq.(197) and (198) we deduce

$c\left(q\right)=\frac{1}{\sqrt{2\pi iB}}\exp\left[\frac{\allowbreak
iA}{2B}q^{2}\right],\text{ }d\left(q\right)=\frac{1}{\sqrt{2\pi
D}}\exp\left[\frac{iCq^{2}}{2D}\right],$ (199)

and

$\left|q\right\rangle_{s,r}=\frac{\pi^{-1/4}}{\sqrt{D+iB}}\exp\left\\{-\frac{A-iC}{D+iB}\frac{q^{2}}{2}+\frac{\sqrt{2}q}{D+iB}a^{\dagger}-\frac{D-iB}{D+iB}\frac{a^{\dagger
2}}{2}\right\\}\left|0\right\rangle,$ (200)

or

$\left|q\right\rangle_{s,r}\equiv\frac{\pi^{-1/4}}{\sqrt{s^{\ast}+\allowbreak
r^{\ast}}}\exp\left\\{-\frac{s^{\ast}-r^{\ast}}{s^{\ast}+r^{\ast}}\frac{q^{2}}{2}+\frac{\sqrt{2}q}{s^{\ast}+r^{\ast}}a^{\dagger}-\frac{s+r}{s^{\ast}+r^{\ast}}\frac{a^{\dagger
2}}{2}\right\\}\left|0\right\rangle.$ (201)

It is easily seen that that $\left|q\right\rangle_{s,r}$ make up a complete
set (so $\left|q\right\rangle_{s,r}$ can be named as the tomography
representation),

$\int_{-\infty}^{\infty}dq\left|q\right\rangle_{s,r}{}_{s,r}\left\langle
q\right|=1.$ (202)

Then according to the Weyl quantization scheme [37]

$H\left(Q,P\right)=\int_{-\infty}^{\infty}dpdq\Delta\left(q,p\right)h\left(q,p\right),$
(203)

where $h\left(q,p\right)$ is the Weyl correspondence of $H\left(Q,P\right),$

$h\left(q,p\right)=2\pi
T_{r}\left[H\left(Q,P\right)\Delta\left(q,p\right)\right],$ (204)

$\Delta\left(q,p\right)$ is the Wigner operator [66, 67],

$\Delta\left(q,p\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}due^{ipu}\left|q+\frac{u}{2}\right\rangle\left\langle
q-\frac{u}{2}\right|,$ (205)

and using (204), (205) and (195) we know that the classical Weyl
correspondence (Weyl image) of the projection operator
$\left|q\right\rangle_{s,rs,r}\left\langle q\right|$ is

$\displaystyle 2\pi
Tr\left[\Delta\left(q^{\prime},p^{\prime}\right)\left|q\right\rangle_{s,rs,r}\left\langle
q\right|\right]$ $\displaystyle=\left.{}_{s,r}\left\langle
q\right|\right.\int_{-\infty}^{\infty}due^{ip^{\prime}u}\left|q^{\prime}+\frac{u}{2}\right\rangle\left\langle
q^{\prime}-\frac{u}{2}\right|\left.q\right\rangle_{s,r}$
$\displaystyle=\frac{1}{2\pi
B}\int_{-\infty}^{\infty}du\exp\left[ip^{\prime}u+\frac{i}{B}u\left(q-Dq^{\prime}\right)\right]$
$\displaystyle=\delta\left[q-\left(Dq^{\prime}-Bp^{\prime}\right)\right],$
(206)

which means

$\left|q\right\rangle_{s,rs,r}\left\langle
q\right|=\int_{-\infty}^{\infty}dq^{\prime}dp^{\prime}\delta\left[q-\left(Dq^{\prime}-Bp^{\prime}\right)\right]\Delta\left(q^{\prime},p^{\prime}\right).$
(207)

Combining Eqs. (187)-(190) together we complete the proof. Therefore, the
probability distribution for the Fresnel quadrature phase is the Radon
transform of WF

$|\left\langle
q\right|F_{1}^{\dagger}\left|\psi\right\rangle|^{2}=|_{s,r}\left\langle
q\right|\left.\psi\right\rangle|^{2}=\int_{-\infty}^{\infty}dq^{\prime}dp^{\prime}\delta\left[q-\left(Dq^{\prime}-Bp^{\prime}\right)\right]\left\langle\psi\right|\Delta\left(q^{\prime},p^{\prime}\right)\left|\psi\right\rangle,$
(208)

so we name $\left|q\right\rangle_{s,rs,r}\left\langle q\right|$ the
tomographic density. Moreover, the tomogram of quantum state
$\left|\psi\right\rangle$ is just the squared modulus of the wave function
${}_{s,r}\left\langle q\right|\left.\psi\right\rangle,$ this new relation
between quantum tomography and optical Fresnel transform may provide
experimentalists to figure out new approach for generating tomography.

The introduction of $\left|q\right\rangle_{s,r}$ also bring convenience to
obtain the inverse of Radon transformation, using (202) we have

$e^{-igQ_{F}}=\int_{-\infty}^{\infty}dq\left|q\right\rangle_{s,r}{}_{s,r}\left\langle
q\right|e^{-igq}=\int_{-\infty}^{\infty}dqdp\Delta\left(q,p\right)e^{-ig\left(Dq-
Bp\right)}.$ (209)

Considering its right hand-side as a Fourier transformation, its reciprocal
transform is

$\displaystyle\Delta\left(q,p\right)$
$\displaystyle=\frac{1}{4\pi^{2}}\int_{-\infty}^{\infty}dq^{\prime}\int_{-\infty}^{\infty}dg^{\prime}|g^{\prime}|\int_{0}^{\pi}d\varphi\left|q^{\prime}\right\rangle_{s,r}{}_{s,r}\left\langle
q^{\prime}\right|$
$\displaystyle\times\exp\left[-ig^{\prime}\left(\frac{q^{\prime}}{\sqrt{D^{2}+B^{2}}}-q\cos\varphi-p\sin\varphi\right)\right],$
(210)

where $g^{\prime}=g\sqrt{D^{2}+B^{2}},$
$\cos\varphi=\frac{D}{\sqrt{D^{2}+B^{2}}},$
$\sin\varphi=\frac{-B}{\sqrt{D^{2}+B^{2}}}.$So once the distribution
$|_{s,r}\left\langle q\right|\left.\psi\right\rangle|^{2}$ are obtained, one
can use the inverse Radon transformation familiar in tomographic imaging to
obtain the Wigner distribution.

By analogy, we can conclude that the $(A,C)$ related Radon transform of
$\Delta\left(q,p\right)$ is just the pure state density operator
$\left|p\right\rangle_{s,rs,r}\left\langle p\right|$ formed with the
eigenstates belonging to the conjugate quadrature of $Q_{F},$

$\displaystyle F_{1}\left|p\right\rangle\left\langle p\right|F_{1}^{\dagger}$
$\displaystyle=\left|p\right\rangle_{s,rs,r}\left\langle
p\right|=\int_{-\infty}^{\infty}dq^{\prime}dp^{\prime}\delta\left[p-\left(Aq^{\prime}-Cp^{\prime}\right)\right]\Delta\left(q^{\prime},p^{\prime}\right),$
$\displaystyle A$
$\displaystyle=\frac{1}{2}\left(s^{\ast}-r^{\ast}+s-r\right),\text{ \
}C=\frac{1}{2i}\left(s-r-s^{\ast}+r^{\ast}\right).$ (211)

Similarly, we find that for the momentum density,

$F_{1}\left|p\right\rangle\left\langle
p\right|F_{1}^{\dagger}=\left|p\right\rangle_{s,rs,r}\left\langle
p\right|=\int_{-\infty}^{\infty}dq^{\prime}dp^{\prime}\delta\left[p-\left(Ap^{\prime}-Cq^{\prime}\right)\right]\Delta\left(q^{\prime},p^{\prime}\right),$
(212)

where

$F_{1}\left|p\right\rangle=\left|p\right\rangle_{s,r}=\frac{\pi^{-1/4}}{\sqrt{A-iC}}\exp\left\\{-\frac{D+iB}{A-iC}\frac{p^{2}}{2}+\frac{\sqrt{2}ip}{A-iC}a^{\dagger}+\frac{A+iC}{A-iC}\frac{a^{\dagger
2}}{2}\right\\}\left|0\right\rangle.$ (213)

As an application of the relation (185), recalling that the $F_{1}(r,s)$ makes
up a faithful representation of the symplectic group [r10], it then follows
from (185) that

$\displaystyle
F_{1}^{\prime}(r^{\prime},s^{\prime})F_{1}(r,s)\left|q\right\rangle\left\langle
q\right|F_{1}^{\dagger}(r,s)F_{1}^{\prime\dagger}(r^{\prime},s^{\prime})=\left|q\right\rangle_{s^{\prime\prime},r^{\prime\prime}\text{
}s^{\prime\prime},r^{\prime\prime}}\left\langle q\right|$
$\displaystyle=\int\int_{-\infty}^{\infty}dq^{\prime}dp^{\prime}\delta\left[q-\left(\left(B^{\prime}C+DD^{\prime}\right)q^{\prime}-\left(AB^{\prime}+BD^{\prime}\right)p^{\prime}\right)\right]\Delta\left(q^{\prime},p^{\prime}\right),$
(214)

In this way a complicated Radon transform of tomography can be viewed as the
sequential operation of two Fresnel transforms. This confirms that the
continuous Radon transformation corresponds to the symplectic group
transformation [63, 64], this is an advantage of introducing the Fresnel
operator. The group property of Fresnel operators help us to analyze
complicated Radon transforms in terms of some sequential Fresnel
transformations. The new relation may provide experimentalists to figure out
new approach for realizing tomography.

###  8.2 Another new theorem to calculating the tomogram

In this subsection, we introduce a new theorem, i.e., the tomogram of a
density operator $\rho$ is equal to the marginal integration of the classical
Weyl correspondence function of $F^{\dagger}\rho F,$ where $F$ is the Fresnel
operator.

Multiplying both sides of Eq. (207) by a density matrix $\rho$ and then
performing the trace, noting the Wigner function
$W(p,q)=\mathtt{Tr}\left[\rho\Delta(p,q)\right],$ one can see

$\displaystyle
Tr\left[{\displaystyle\iint\nolimits_{-\infty}^{\infty}}dq\acute{}dp\acute{}\delta\left[q-\left(Dq\acute{}-Bp\acute{}\right)\right]\Delta\left(q\acute{},p\acute{}\right)\rho\right]$
$\displaystyle=Tr\left(\left|q\right\rangle_{s,rs,r}\left\langle
q\right|\rho\right)=_{s,r}\left\langle
q\right|\rho\left|q\right\rangle_{s,r}=\left\langle q\right|F^{\dagger}\rho
F\left|q\right\rangle$
$\displaystyle={\displaystyle\iint\nolimits_{-\infty}^{\infty}}dq\acute{}dp\acute{}\delta\left[q-\left(Dq\acute{}-Bp\acute{}\right)\right]W(p,q).$
(215)

The right hand side of Eq. (215) is commonly defined as the tomogram of
quantum states in $(B,D)$ direction, so in our view the calculation of
tomogram in $(B,D)$ direction is ascribed to calculating

$\left\langle q\right|F^{\dagger}\rho F\left|q\right\rangle\equiv\Xi.$ (216)

This is a concise and neat formula. Similarly, the tomogram in $(A,C)$
direction is ascribed to $\left\langle p\right|F^{\dagger}\rho
F\left|p\right\rangle$.

According to the Weyl correspondence rule

$H\left(X,P\right)=\iint_{-\infty}^{\infty}dpdx\mathfrak{h}(p,x)\Delta(p,x),$
(217)

and the Weyl ordering form of $\Delta(p,q)$

$\Delta(p,q)=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q-Q\right)\delta\left(p-P\right)\genfrac{}{}{0.0pt}{}{:}{:},$
(218)

where the symbol$\genfrac{}{}{0.0pt}{}{:}{:}\ \genfrac{}{}{0.0pt}{}{:}{:}$
denotes Weyl ordering, the classical correspondence of a Weyl ordered operator
$\genfrac{}{}{0.0pt}{}{:}{:}\mathfrak{h}(Q,P)\genfrac{}{}{0.0pt}{}{:}{:}$ is
obtained just by replacing $Q\rightarrow q,P\rightarrow p$ in $h,$ i.e.,

$\genfrac{}{}{0.0pt}{}{:}{:}\mathfrak{h}(Q,P)\genfrac{}{}{0.0pt}{}{:}{:}=\iint_{-\infty}^{\infty}dpdq\mathfrak{h}(p,q)\Delta(p,q),$
(219)

Let the classical Weyl correspondence of $F^{\dagger}\rho F$ be $h(p,q)$

$F^{\dagger}\rho F=\iint_{-\infty}^{\infty}dpdqh(p,q)\Delta(p,q),$

then using (216) and (205) we have

$\displaystyle\Xi$ $\displaystyle=\left\langle q\right|F^{\dagger}\rho
F\left|q\right\rangle$ $\displaystyle=\left\langle
q\right|{\displaystyle\iint}dpdq\acute{}h(p,q\acute{})\Delta\left(p,q\acute{}\right)\left|q\right\rangle$
$\displaystyle={\displaystyle\iint}dpdq\acute{}h(p,q\acute{})\int_{-\infty}^{+\infty}\frac{dv}{2\pi}e^{ipv}\left\langle
x\right.\left.q\acute{}+\frac{v}{2}\right\rangle\left\langle
q\acute{}-\frac{v}{2}\right.\left.q\right\rangle$
$\displaystyle={\displaystyle\iint}dpdq\acute{}h(p,q\acute{})\int_{-\infty}^{+\infty}\frac{dv}{2\pi}e^{ipv}\delta\left(q\acute{}-q+\frac{v}{2}\right)\delta\left(q\acute{}-q-\frac{v}{2}\right)$
$\displaystyle=\frac{1}{\pi}{\displaystyle\iint}dpdq\acute{}h(p,q\acute{})e^{i2p\left(q\acute{}-q\right)}\delta\left(2q\acute{}-2q\right)=\int_{-\infty}^{\infty}\frac{dp}{2\pi}h(p,q).$
(220)

Thus we reach a theorem:

The tomogram of a density operator $\rho$ is equal to the marginal integration
of the classical Weyl correspondence $h(p,q)$ of $F^{\dagger}\rho F,$ where
$F$ is the Fresnel operator, expressed by

$\mathtt{Tr}\left[\rho\left|q\right\rangle_{s,rs,r}\left\langle
q\right|\right]=\int_{-\infty}^{\infty}\frac{dp}{2\pi}h(p,q),$ (221)

or

$\mathtt{Tr}\left[\rho\left|p\right\rangle_{s,rs,r}\left\langle
p\right|\right]=\int_{-\infty}^{\infty}\frac{dx}{2\pi}h(p,q).$ (222)

In this way the relationship between tomogram of a density operator $\rho$ and
the Fresnel transformed $\rho^{\prime}$s classical Weyl function is
established.

##  9 Two-mode GFO and Its Application

For two-dimensional optical Fresnel transforms (see (5)) in the $x-y$ plane
one may naturally think that the 2-mode GFO is just the direct product of two
independent 1-mode GFOs but with the same $(A,B,C,D)$ matrix. However, here we
present another 2-mode Fresnel operator which can not only lead to the usual
2-dimensional optical Fresnel transforms in some appropriate quantum
mechanical representations, but also provide us with some new classical
transformations (we name them entangled Fresnel transformations).

###  9.1 Two-mode GFO gained via coherent state representation

Similar in spirit to the single-mode case, we introduce the two-mode GFO
$F_{2}\left(r,s\right)$ through the following 2-mode coherent state
representation [50]

$F_{2}\left(r,s\right)=s\int\frac{d^{2}z_{1}d^{2}z_{2}}{\pi^{2}}\left|sz_{1}+rz_{2}^{\ast},rz_{1}^{\ast}+sz_{2}\right\rangle\left\langle
z_{1},z_{2}\right|,$ (223)

which indicates that $F_{2}\left(r,s\right)$ is a mapping of classical
sympletic transform
$\left(z_{1},z_{2}\right)\rightarrow\left(sz_{1}+rz_{2}^{\ast},rz_{1}^{\ast}+sz_{2}\right)$
in phase space. Concretely, the ket in (223) is

$\left|sz_{1}+rz_{2}^{\ast},rz_{1}^{\ast}+sz_{2}\right\rangle\equiv\left|sz_{1}+rz_{2}^{\ast}\right\rangle_{1}\otimes\left|rz_{1}^{\ast}+sz_{2}\right\rangle_{2},\text{
}ss^{\ast}-rr^{\ast}=1,$ (224)

$s$ and $r$ are complex and satisfy the unimodularity condition. Using the
IWOP technique we perform the integral in (223) and obtain

$\displaystyle F_{2}\left(r,s\right)$
$\displaystyle=s\int\frac{1}{\pi^{2}}d^{2}z_{1}d^{2}z_{2}\colon\exp[-|s|^{2}\left(|z_{1}|^{2}+|z_{2}|^{2}\right)-r^{\ast}sz_{1}z_{2}-rs^{\ast}z_{1}^{\ast}z_{2}^{\ast}$
$\displaystyle+\left(sz_{1}+rz_{2}^{\ast}\right)a_{1}^{\dagger}+\left(rz_{1}^{\ast}+sz_{2}\right)a_{2}^{\dagger}+z_{1}^{\ast}a_{1}+z_{2}^{\ast}a_{2}-a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}]\colon$
$\displaystyle=\frac{1}{s^{\ast}}\exp\left(\frac{r}{s^{\ast}}a_{1}^{\dagger}a_{2}^{\dagger}\right)\colon\exp\left[\left(\frac{1}{s^{\ast}}-1\right)\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\right]\colon\exp\left(-\frac{r^{\ast}}{s^{\ast}}a_{1}a_{2}\right)$
$\displaystyle=\exp\left(\frac{r}{s^{\ast}}a_{1}^{\dagger}a_{2}^{\dagger}\right)\exp[\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}+1\right)\ln\left(s^{\ast}\right)^{-1}]\exp\left(-\frac{r^{\ast}}{s^{\ast}}a_{1}a_{2}\right).$
(225)

Thus $F_{2}\left(r,s\right)$ induces the transform

$F_{2}\left(r,s\right)a_{1}F_{2}^{-1}\left(r,s\right)=s^{\ast}a_{1}-ra_{2}^{\dagger},F_{2}\left(r,s\right)a_{2}F_{2}^{-1}\left(r,s\right)=s^{\ast}a_{2}-ra_{1}^{\dagger}.$
(226)

and $F_{2}$ is actually a general 2-mode squeezing operator. Recall that (37)
implies the intrinsic relation between the EPR entangled state and the two-
mode squeezed state, which has physical implementation, i.e. in the output of
a parametric down-conversion the idler-mode and the signal-mode constitute a
two-mode squeezed state, meanwhile are entangled with each other in frequency
domain, we naturally select the entangled state representation to relate
$F_{2}\left(r,s\right)$ to two-dimensional GFT. Letting
$\left|g\right\rangle=F_{2}\left(r,s\right)\left|f\right\rangle,$ and then
projecting $\left|\psi\right\rangle$ onto the entangled state
$\left\langle\eta^{\prime}\right|$ defined by (29) and using the completeness
relation (31) of $\left|\eta\right\rangle$, we obtain

$\displaystyle g\left(\eta^{\prime}\right)$
$\displaystyle\equiv\left\langle\eta^{\prime}\right|\left.g\right\rangle=\left\langle\eta^{\prime}\right|F_{2}\left(r,s\right)\left|f\right\rangle$
$\displaystyle=\int\frac{d^{2}\eta}{\pi}\left\langle\eta^{\prime}\right|F_{2}\left(r,s\right)\left|\eta\right\rangle\left\langle\eta\right.\left|f\right\rangle\equiv\int
d^{2}\eta\mathcal{K}_{2}^{\left(r,s\right)}\left(\eta^{\prime},\eta\right)f\left(\eta\right).$
(227)

Then using the overcompleteness relation of the coherent state, and

$\displaystyle\left\langle
z_{1}^{\prime},z_{2}^{\prime}\right|F_{2}\left(r,s\right)\left|z_{1},z_{2}\right\rangle$
$\displaystyle=\frac{1}{s^{\ast}}\exp\left\\{-\frac{1}{2}(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{1}^{\prime}\right|^{2}+\left|z_{2}^{\prime}\right|^{2})\right.$
$\displaystyle\left.+\frac{r}{s^{\ast}}z_{1}^{\prime\ast}z_{2}^{\prime\ast}-\frac{r^{\ast}}{s^{\ast}}z_{1}z_{2}+\frac{1}{s^{\ast}}\left(z_{1}^{\prime\ast}z_{1}+z_{2}^{\prime\ast}z_{2}\right)\right\\},$

we can calculate the integral kernel

$\displaystyle\mathcal{K}_{2}^{\left(r,s\right)}\left(\eta^{\prime},\eta\right)$
$\displaystyle=\frac{1}{\pi}\left\langle\eta^{\prime}\right|F_{2}\left(r,s\right)\left|\eta\right\rangle$
$\displaystyle=\int\frac{d^{2}z_{1}d^{2}z_{2}d^{2}z_{1}^{\prime}d^{2}z_{2}^{\prime}}{\pi^{5}}\left\langle\eta^{\prime}\right|\left.z_{1}^{\prime},z_{2}^{\prime}\right\rangle\left\langle
z_{1}^{\prime},z_{2}^{\prime}\right|F_{2}\left(r,s\right)\left|z_{1},z_{2}\right\rangle\left\langle
z_{1},z_{2}\right.\left|\eta\right\rangle$
$\displaystyle=\frac{1}{s^{\ast}}\int\frac{d^{2}z_{1}d^{2}z_{2}d^{2}z_{1}^{\prime}d^{2}z_{2}^{\prime}}{\pi^{5}}\exp\left[-\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{1}^{\prime}\right|^{2}+\left|z_{2}^{\prime}\right|^{2}\right)-\frac{1}{2}\left(\left|\eta^{\prime}\right|^{2}+\left|\eta\right|^{2}\right)\right]$
$\displaystyle\times\exp\left[-\frac{r^{\ast}}{s^{\ast}}z_{1}z_{2}+z_{1}^{\ast}z_{2}^{\ast}+\eta
z_{1}^{\ast}+\frac{1}{s^{\ast}}\left(z_{1}^{\prime\ast}z_{1}+z_{2}^{\prime\ast}z_{2}\right)+\frac{r}{s^{\ast}}z_{1}^{\prime\ast}z_{2}^{\prime\ast}+z_{1}^{\prime}z_{2}^{\prime}+\eta^{\prime\ast}z_{1}^{\prime}-\eta^{\prime}z_{2}^{\prime}-\eta^{\ast}z_{2}^{\ast}\right]$
$\displaystyle=\frac{1}{\left(-r-s+r^{\ast}+s^{\ast}\right)\pi}\exp\left[\frac{\left(-s+r^{\ast}\right)\left|\eta\right|^{2}-\left(r+s\right)\left|\eta^{\prime}\right|^{2}+\eta\eta^{\prime\ast}+\eta^{\ast}\eta^{\prime}}{-r-s+r^{\ast}+s^{\ast}}-\frac{1}{2}\left(\left|\eta^{\prime}\right|^{2}+\left|\eta\right|^{2}\right)\right].$
(228)

Using the relation between $s,r$ and $\left(A,B,C,D\right)$ in Eq.(85) Eq.
(228) becomes

$\mathcal{K}_{2}^{\left(r,s\right)}\left(\eta^{\prime},\eta\right)=\frac{1}{2iB\pi}\exp\left[\frac{i}{2B}\left(A\left|\eta\right|^{2}-\left(\eta\eta^{\prime\ast}+\eta^{\ast}\eta^{\prime}\right)+D\left|\eta^{\prime}\right|^{2}\right)\right]\equiv\mathcal{K}_{2}^{M}\left(\eta^{\prime},\eta\right),$
(229)

where the superscript $M$ only means the parameters of $K_{2}^{M}$ are
$\left[A,B;C,D\right]$, and the subscript $2$ means the two-dimensional
kernel. Eq. (229) has the similar form as (12) except for its complex form.
Taking $\eta_{1}=x_{1},$ $\eta_{2}=x_{2}$ and
$\eta_{1}^{\prime}=x_{1}^{\prime}$, $\eta_{2}^{\prime}=x_{2}^{\prime}$, we
have

$\mathcal{K}_{2}^{M}\left(\eta^{\prime},\eta\right)=\mathcal{K}_{2}^{M}\left(x_{1}^{\prime},x_{2}^{\prime};x_{1},x_{2}\right)=\mathcal{K}_{1}^{M}\left(x_{1},x_{1}^{\prime}\right)\otimes\mathcal{K}_{1}^{M}\left(x_{2},x_{2}^{\prime}\right).$
(230)

This shows that $F_{2}\left(r,s\right)$ is really the counterpart of the
2-dimensional GFT.

If taking the matrix element of $F_{2}\left(r,s\right)$ in the
$\left|\xi\right\rangle$ representation which is conjugate to
$\left|\eta\right\rangle$, we obtain the 2-dimensional GFT in its ‘frequency
domain’, i.e.,

$\displaystyle\left\langle\xi^{\prime}\right|F_{2}\left(r,s\right)\left|\xi\right\rangle$
$\displaystyle=\int\frac{d^{2}\eta^{\prime}d^{2}\eta}{\pi^{2}}\left\langle\xi^{\prime}\right|\left.\eta^{\prime}\right\rangle\left\langle\sigma\right|F_{2}\left(r,s\right)\left|\eta\right\rangle\left\langle\eta\right|\left.\xi\right\rangle$
$\displaystyle=\frac{1}{8iB\pi}\int\frac{d^{2}\eta^{\prime}d^{2}\eta}{\pi^{2}}\mathcal{K}_{2}^{\left(r,s\right)}\left(\eta^{\prime},\eta\right)\exp\left(\frac{\xi^{\prime\ast}\eta^{\prime}-\xi^{\prime}\eta^{\prime\ast}+\xi\eta^{\ast}-\xi^{\ast}\eta}{2}\right)$
$\displaystyle=\frac{1}{2i\left(-C\right)\pi}\exp\left[\frac{i}{2\left(-C\right)}\left(D\left|\xi\right|^{2}+A\left|\xi^{\prime}\right|^{2}-\xi^{\prime\ast}\xi-\xi^{\prime}\xi^{\ast}\right)\right]\equiv\mathcal{K}_{2}^{N}\left(\xi^{\prime},\xi\right),$
(231)

where the superscript $N$ means that this transform kernel corresponds to the
parameter matrix $N=\left[D,-C,-B,A\right]$.

The two-mode GFO also abides by the group multiplication rule. Using the IWOP
technique and (223) we obtain

$\displaystyle F_{2}\left(r,s\right)F_{2}\left(r^{\prime},s^{\prime}\right)$
$\displaystyle=ss^{\prime}\int\frac{d^{2}z_{1}d^{2}z_{2}d^{2}z_{1}^{\prime}d^{2}z_{2}^{\prime}}{\pi^{4}}\colon\exp\\{-|s|^{2}\left(|z_{1}|^{2}+|z_{2}|^{2}\right)-r^{\ast}sz_{1}z_{2}$
$\displaystyle-
rs^{\ast}z_{1}^{\ast}z_{2}^{\ast}-\frac{1}{2}[|z_{1}^{\prime}|^{2}+|z_{2}^{\prime}|^{2}+|s^{\prime}z_{1}^{\prime}+r^{\prime}z_{2}^{\prime\ast}|^{2}+|r^{\prime}z_{1}^{\prime\ast}+s^{\prime}z_{2}^{\prime}|^{2}]$
$\displaystyle+\left(sz_{1}+rz_{2}^{\ast}\right)a_{1}^{\dagger}+\left(rz_{1}^{\ast}+sz_{2}\right)a_{2}^{\dagger}+z_{1}^{\prime\ast}a_{1}+z_{2}^{\prime\ast}a_{2}$
$\displaystyle+z_{1}^{\ast}\left(s^{\prime}z_{1}^{\prime}+r^{\prime}z_{2}^{\prime\ast}\right)+z_{2}^{\ast}\left(r^{\prime}z_{1}^{\prime\ast}+s^{\prime}z_{2}^{\prime}\right)-a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}\\}\colon$
$\displaystyle=\frac{1}{s^{\prime\prime\ast}}\exp\left(\frac{r^{\prime\prime}}{2s^{\prime\prime\ast}}a_{1}^{\dagger}a_{2}^{\dagger}\right)\colon\exp\left\\{\left(\frac{1}{s^{\prime\prime\ast}}-1\right)\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)\right\\}\colon\exp\left(-\frac{r^{\prime\prime\ast}}{2s^{\prime\prime\ast}}a_{1}a_{2}\right)$
$\displaystyle=F_{2}\left(r^{\prime\prime},s^{\prime\prime}\right),$ (232)

where $\left(r^{\prime\prime},s^{\prime\prime}\right)$ are given by Eq.(106)
or (107). Therefore, (232) is a loyal representation of the multiplication
rule for ray transfer matrices in the sense of Matrix Optics.

###  9.2 Quantum Optical ABCD Law for two-mode GFO

Next we extended quantum optical ABCD Law to two-mode case. Operating with
$F_{2}(r,s)$ on two-mode number state $\left|m,n\right\rangle$ and using the
overlap between coherent state and number state, i.e.

$\left\langle z_{1},z_{2}\right.\left|m,n\right\rangle=\frac{z_{1}^{\ast
m}z_{2}^{\ast
n}}{\sqrt{m!n!}}\exp\left[-\frac{1}{2}\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right)\right],$
(233)

and the integral formula [57]

$H_{m,n}\left(\xi,\eta\right)=(-1)^{n}e^{\xi\eta}\int\frac{d^{2}z}{\pi}z^{n}z^{\ast
m}e^{-\left|z\right|^{2}+\xi z-\eta z^{\ast}},$ (234)

we can calculate

$\displaystyle F_{2}(r,s)\left|m,n\right\rangle$
$\displaystyle=s\int\frac{d^{2}z_{1}d^{2}z_{2}}{\pi^{2}}\left|sz_{1}+rz_{2}^{\ast},rz_{1}^{\ast}+sz_{2}\right\rangle\left\langle
z_{1},z_{2}\right.\left|m,n\right\rangle$
$\displaystyle=\frac{s}{\sqrt{m!n!}}\int\frac{d^{2}z_{1}d^{2}z_{2}}{\pi^{2}}z_{1}^{\ast
m}z_{2}^{\ast
n}\exp\left[-\left|s\right|^{2}\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right)\right]$
$\displaystyle\times\exp\left[-sr^{\ast}z_{1}z_{2}-rs^{\ast}z_{1}^{\ast}z_{2}^{\ast}+\left(sz_{1}+rz_{2}^{\ast}\right)a_{1}^{\dagger}+\left(rz_{1}^{\ast}+sz_{2}\right)a_{2}^{\dagger}\right]\left|00\right\rangle$
$\displaystyle=\frac{s}{\left|s\right|^{2m+2}\sqrt{m!n!}}\int\frac{d^{2}z_{2}}{\pi}z_{2}^{\ast
n}\left(sa_{1}^{\dagger}-sr^{\ast}z_{2}\right)^{m}\exp\left(-\left|z_{2}\right|^{2}+\frac{1}{s^{\ast}}a_{2}^{\dagger}z_{2}+\frac{ra_{1}^{\dagger}a_{2}^{\dagger}}{s^{\ast}}\right)\left|00\right\rangle$
$\displaystyle=\frac{r^{\ast n}}{s^{\ast
n+1}\sqrt{m!n!}}H_{m,n}\left[\frac{a_{1}^{\dagger}}{s^{\ast}},\frac{a_{2}^{\dagger}}{r^{\ast}}\right]\exp\left(\frac{ra_{1}^{\dagger}a_{2}^{\dagger}}{s^{\ast}}\right)\left|00\right\rangle,$
(235)

where $H_{m,n}\left(\epsilon,\varepsilon\right)$ is the two variables Hermite
polynomial [63, 64], shown in (78) and (79).

Using Eqs.(85) and (118), we recast Eq.(235) into

$\displaystyle F_{2}(r,s)\left|m,n\right\rangle$
$\displaystyle=\frac{-2/\left(C+iD\right)}{\left(q_{1}+i\right)\sqrt{m!n!}}\left(-\frac{q_{1}^{\ast}+i}{q_{1}+i}\frac{C-iD}{C+iD}\right)^{n}$
$\displaystyle\times