Optical Four-Wave Mixing Operator, Fresnel Operators and Three-Mode Entangled State Representation

We analyse the optical four-wave mixing operator S and relate it to the two-mode Fresnel operator. It is shown that the direct product of the two-mode Fresnel operator and the single-mode Fresnel operator has a natural representation on the basis of a three-mode entangled state, which is constructed by S and a beam splitter transform.     

Fresnel-Transform's Quantum Correspondence and Quantum Optical ABCD Law

Corresponding to the Fresnel transform there exists a unitary operator in quantum optics theory, which could be known the Fresnel operator (FO). We show that the multiplication rule of the FO naturally leads to the quantum optical ABCD law. The canonical operator methods as mapping of ray-transferABCD matrix is explicitly shown by the normally ordered expansion of the FO through the coherent state representation and the technique of integration within an ordered product of operators. We show that time evolution of the damping oscillator embodies the quantum optical ABCD law.     

New decomposition of the Fresnel operator corresponding to the optical transformation in AB CD-systems

By virtue of the coherent state representation of the newly introduced Fresnel operator and its group product property we obtain new decomposition of the Fresnel operator as the product of the quadratic phase operator, the squeezing operator, and the fractional Fourier transformation operator, which in turn sheds light on the matrix optics design of ABCD-systems The new decomposition for the two-mode Fresnel operator is also obtained by the use of entangled state representation.     

Weyl Ordering of Two-Mode Fresnel Operator and Its Applications

By using the Weyl ordering operator formula and the Weyl transformation rule, we derive Weyl ordering of the two-mode Fresnel operator, and then obtain its matrix element in the coordinate representation, which is the integral kernel of the generalized two-mode Fresnel transformation in classical optics. By means of the optical operator method, we obtain a decomposition of the two-mode Fresnel operator’s Weyl ordering and an operator identity.     

Entangled symplectic wavelet transformation

The symplectic wavelet transformation proposed in Opt. Lett. 31, 3432 (2006), which is related to the optical Fresnel transform in the quantum optics version, is developed into an entangled symplectic wavelet transformation (ESWT) after pointing out the contrast between the single-mode Fresnel operator and the entangled Fresnel operator. The ESWT possesses well-behaved properties and corresponds to the entangled Fresnel transform [Phys. Lett. A334, 132 (2005)].     

Fresnel Operator for Deriving the Propagator of 2D Harmonic Oscillator with Cross Coupling

Using the identity of operator decomposition we obtain a normal ordered form of the time-evolution operator for cross coupling quantum harmonic oscillator Hamiltonian system in two dimensions, which is just a special two-mode Fresnel operator. The Feynman propagator for the Hamiltonian system is found by a direct calculation by means of the method deriving the matrix element of two-mode Fresnel operator in the entangled state representation. The technique of integration within an ordered product (IWOP) of operators is employed to derive the matrix elements of the operator in the coherent state and the entangled state representations.     

The S-ordered Operator Expansions of One-mode and Two-mode Fresnel Operators and their Applications

By using the technique of integration within the s-ordered product of operators (IWSOP), we first deduce the s-ordered expansion of the one-mode and two-mode Fresnel operators. Employing the s-ordered operator expansion formula, the matrix elements of one-mode and two-mode Fresnel operator in the number state representation are also obtained, respectively.     

Optical Operator Method Studied via Fresnel Operator Decomposition and Coherent State Representation

We find that the mapping from classical optical transformations to the optical operator method can be realized by using the coherent state representation and the technique of integration within an ordered product of operators. The optical Fresnel operator derived in (Commun. Theor. Phys. (Beijing, China) 38 (2002) 147) can unify those frequently used optical operators. Various decompositions of Fresnel operator into the exponential canonical operators are obtained.     

经典菲涅耳变换的量子力学对应

介绍经典菲涅耳变换的量子力学对应,它是相干态在相空间中的代表点做辛运动所对应的量子菲涅耳算符.     

New approach to Q-P （P-Q） ordering of quantum mechanical operators and its applications

In this paper, we introduce a new way to obtain the Q-P （P-Q） ordering of quantum mechanical operators, i.e., from the classical correspondence of Q-P （P-Q） ordered operators by replacing q and p with coordinate and momentum operators, respectively. Some operator identities are derived concisely. As for its applications, the single （two-） mode squeezed operators and Fresnel operator are examined. It is shown that the classical correspondence of Fresnel operator’s Q-P （P-Q） ordering is just the integration kernel of Fresnel transformation. In addition, a new photo-counting formula is constructed by the Q-P ordering of operators.     

Numerical analysis for finite Fresnel transform

Optical Review - The Fresnel transform is a bounded, linear, additive, and unitary operator in Hilbert space and is applied to many applications. In this study, a sampling theorem for a Fresnel...     

ABCD rule for Gaussian beam propagation in the context of quantum optics derived by the IWOP technique

The development of technique of integration within an ordered product (IWOP) of operators extends the Newton-Leibniz integration rule, originally applying to permutable functions, to the non-commutative quantum mechanical operators composed of Dirac’s ket-bra, which enables us to obtain the images of directly mapping symplectic transformation in classical phase space parameterized by [A, B; C, D] into quantum mechanical operator through the coherent state representation, we call them the generalized Fresnel operators (GFO) since they correspond to Fresnel transforms in Fourier optics. Based on GFO we find the ABCD rule for Gaussian beam propagation in the context of quantum optics (both in one-mode and two-mode cases) whose classical correspondence is just the ABCD rule in matrix optics. The entangled state representation is used in discussing the two-mode case.     

Weyl Ordering Operator Formula Derived by IWOP Technique and Its Application for Fresnel Operator

Based on the technique of integration within an ordered product of operators, the Weyl ordering operator formula is derived and the Fresnel operators＇ Weyl ordering is also obtained, which together with the Weyl transformation can immediately lead to Eresnel transformation kernel in classical optics.     

A generalized Collins formula derived by virtue of the displacement-squeezing related squeezed coherent state representation

Based on the displacement-squeezing related squeezed coherent state representation ≤ft\vert z\right\rangle _{g} and using the technique of integration within an ordered product of operators, this paper finds a generalized Fresnel operator, whose matrix element in the coordinate representation leads to a generalized Collins formula (Huygens--Fresnel integration transformation describing optical diffraction). The generalized Fresnel operator is derived by a quantum mechanical mapping from z to sz-rz^{\ast } in the % ≤ft\vert z\right\rangle _{g} representation, while ≤ft\vert z\right\rangle _{g} in phase space is graphically denoted by an ellipse.     

A new theorem relating quantum tomogram to the Fresnel operator

According to Fan-Hu's formalism (Fan Hong-Yi and Hu Li-Yun 2009 Opt. Commun. bf282 3734) that the tomogram of quantum states can be considered as the module-square of the state wave function in the intermediate coordinate-momentum representation which is just the eigenvector of the Fresnel quadrature phase, we derive a new theorem for calculating quantum tomogram of density operator, i.e., the tomogram of a density operator ρ is equal to the marginal integration of the classical Weyl correspondence function of F⁺ρF, where F is the Fresnel operator. Applications of this theorem to evaluating the tomogram of optical chaotic field and squeezed chaotic optical field are presented.     

Time Evolution Caused by Hamiltonian Composed of Quadratic Combination of Canonical Operators and Time-Dependent Two-Mode Fresnel Operator

We show that the time-dependent two-mode Fresnel operator is just the time-evolutional unitary operator governed by the Hamiltonian composed of quadratic combination of canonical operators in the way of exhibiting SU（1,1） algebra. This is an approach for obtaining the time-dependent Hamiltonian from the preassigned time evolution in classical phase space, an approach which is in contrast to Lewis-Riesenfeld＇s invariant operator theory of treating timedependent harmonic oscillators.     

Quantum optical ABCD theorem in two-mode case

By introducing the entangled Fresnel operator （EFO） this paper demonstrates that there exists ABCD theorem for two-mode entangled case in quantum optics. The canonical operator method as mapping of ray-transfer ABCD matrix is explicitly shown by EFO＇s normally ordered expansion through the coherent state representation and the technique of integration within an ordered product of operators.     

Optical Fresnel Transform as a Correspondence of the SU(1,1) Squeezing Operator Composed of Quadratic Combination of Canonical Operators and the Entangled State

We study optical Fresnel transforms by finding the appropriate quantum mechanical SU(1,1) squeezing operators which are composed of quadratic combination of canonical operators. In one-mode case, the squeezing operator's matrix element in the coordinate basis is just the kernel of one-dimensional generalized Fresnel transform (GFT); while in two-mode case, the matrix element of the squeezing operator in the entangled state basis leads to the two-dimensional GFT kernel. The work links optical transforms in wave optics to generalized squeezing transforms in quantum optics.     

Optical scaled Fresnel--Fourier transform obtained via intermediate coordinate-- momentum representation

Using the intermediate coordinate--momentum representation |x>_s,r, we introduce a new Hadamard transform. It is found that the operator U corresponding to this transform can be considered as a combination of the Fresnel operator F(r,s) and the Fourier transform operator F by decomposing U. We also find that the matrix element _s,r< x| U|f> just corresponds to an optical scaled Fresnel--Fourier transform.     

Weyl Ordering Operator Formula Derived by IWOP Technique and ItsApplication for Fresnel Operator

Based on the technique of integration within an ordered product ofoperators, the Weyl ordering operator formula is derived and the Fresneloperators' Weyl ordering is also obtained, which together with the Weyltransformation can immediately lead to Fresnel transformation kernel inclassical optics.